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Eric Weisstein's World of Mathematics : Has Definitions and References For all terms below (Figurate Number, Pentagonal Number, Pyramidal, etc.), exceptJonathan Numbers, which are classes of sets of "Iterated Figurate Numbers" as defined in several of Jonathan Vos Post's publications as cited elsewhere on this page. "Constructable Numbers" are Polygonal, but higher orders of them are omitted below [i.e. 257-gon(2), 65537-gon(100)]...Which N-gons up to N-gon(R) are in this table?Also see the companion table for higher-dimensional numbers:## Table of Polytope Numbers, Sorted, Through 1,000,000

I created this table of 4-D, 5-D, 6-D, 7-D, 8-D, 9-D, and 10-D Polytope Numbers as a useful and fun reference list while working on several of the Number Theory papers listed below. It goes far beyond 1,000,000 for some sequences.## 1:

many sequences of Figurate Numbers start with 1. Hence, among other things, 1 is the first of the Triangular, Square, Pentagonal, Hexagonal, Heptagonal, Octagonal, Nonagonal, Decagonal and other Polygonal Numbers. It is also, after 0, the first Square, Cube, Biquadratic, Fifth Power, and so forth. It is also the first of many Polyhedral Numbers such as the Tetrahedron, Square Pyramid, Pentagonal Pyramid, Octahedron, Dodecahedron, and Icosahedron. It is also the first of an infinite number of multidimensional Polytope Numbers such as the Pentatope (4-Simplex or Hypertetrahedron), Hyperoctahedron, 24-cell, Hyperdodecahedron, and Hypericosahedron -- these Polytope numbers are the subject of a recent paper by this author, and will be given their own web page soon.## 2:

The 2-gon (or digon) is a degenerate polygon, with two edges connecting two vertices. On a Euclidean Plane, the two edges are exactly overlapping. On the surface of a sphere, however, we see digons when we look at two lines of longitude meeting at the North Pole and South Pole, but otherwise taking different paths. 2 is the only even prime; a regular polygon may always be constructed with compass and straightedge if that number of sides is 2 times as many as some other constructable regular polygon, by bisecting each side## 3:

the first nontrivial Triangular Number, or the second total T(2); also a Constructable Number because a Fermat Prime## 4:

the first nontrivial Square Number, or the second total S(2); also the first nontrivial Tetrahedral Number Tet(2)## 5:

the first nontrivial Pentagonal Number, or the second total P(2); also the first nontrivial Pentatope Number Ptop(2), also the first nontrivial Square Pyramidal Number SPyr(2), also a Fermat Prime, and hence a regular Pentagon can be constructed by compass and straightedge (a Constructable Number); also the first nontrvial Centered Square Number CS(2)## 6:

the first nontrivial Hexagonal Number, or the second total H(2) also a Triangular Number T(3), also the first nontrivial Jonathan Number T(T(2)), also the first Semiprime Triangular Number, also the first Semiprime Hexagonal Number, also the first nontrivial Centered Pentagonal Number CP(2), also the first nontrivial Octahedral Number Octh(2), also the first nontrivial Pentagonal Pyramidal Number PPyr(2), also a Constructable Number because 2x3 (power of two times product of Fermat Primes)## 7:

the first nontrivial Heptagonal Number, or the second total Hep(2), also the first nontrivial Centered Hexagonal Number CH(2), also the first nontrivial Hexagonal Pyramidal Number HPyr(2)## 8:

the first nontrivial Cube, C(2), also the first nontrivial Octagonal Number, or the second total O(2); also the first nontrivial Heptagonal Pyramidal Number HepPyr(2)## 9:

Square Number S(3), also the first nontrivial Nonagonal Number, or the second total N(2), also the first Semiprime Nonagonal Number (3 x 3), also the first nontrivial Centered Cube Number CC(2)## 10:

the first nontrivial Decagonal Number, or the second total D(2), also the first Semiprime Decagonal Number (2 x 5), also a Triangular Number T(4), also the 2nd Semiprime Triangular Number, also a Centered Triangle CT(3), also the Tetrahedral Number Tet(3) a product of a power of two and a Fermat Prime, 2x5, and hence a regular 10-gon can be constructed by compass and straightedge, as Gauss first discovered## 11:

the first nontrivial Hendecagonal Number, or 11-gonal Number, or the second total Hen(2)## 12:

a Pentagonal P(3), Dodecagonal Dod(1), Hendecagonal Pyramidal Number, also a Constructable Number because (2^2)x3 (power of two times product of Fermat Primes)## 13:

a Centered Square Number CS(3) also the first nontrivial 13-gonal Number, 13gon(2)## 14:

the Square Pyramidal Number SPyr(3), also the first nontrivial Stella Octangula Number StOct(2), also the first nontrivial 14-gonal Number, 14gon(2)## 15:

Triangular Number T(5), also Hexagonal Number H(3), also the 3rd Semiprime Triangular Number, also the 2nd Semiprime Hexagonal Number, also the first nontrivial Rhombic Dodecahedral Number RhoDod(2), also the Pentatope Number Ptop(3) also the first nontrivial 15-gonal Number, 15gon(2) also a product of Fermat Primes, 3x5, and hence a regular 15-gon can be constructed by compass and straightedge, as Gauss first discovered## 16:

Square Number S(4), also the first nontrivial Biquadratic Number (4th power) 2^4 also the Centered Pentagonal Number CP(3), also the Truncated Tetrahedral Number TruTet(2), also the first nontrivial 16-gonal number, 16gon(2), also a contructable number because a power of 2 (we won't mention that again for higher powers of 2)## 17:

a Fermat Prime, and hence a regular 17-gon can be constructed by compass and straightedge, as Gauss first discovered; also the first nontrivial 17-gonal Number, 17gon(2)## 18:

Heptagonal Number Hep(3), also the Pentagonal Pyramidal Number PPyr(3), also the first nontrivial 18-gonal Number, 18gon(2)## 19:

Octahedral Number Octh(3), also a Centered Triangle CT(4), also Centered Hexagonal Number CH(3), also the first nontrivial 19-gonal Number, 19gon(2)## 20:

the Tetrahedral Number Tet(4), also the the Tetrahedral Jonathan Number Tet(Tet(2)), a Constructable Number because (2^2)x5 (power of two times product of Fermat Primes), also the first nontrivial 20-gonal Number, 20gon(2)## 21:

Triangular Number T(6), thus the Jonathan Number T(T(3))=T(T(T(2))), also the 4th Semiprime Triangular Number, also the Octagonal Number O(3), also the first Semiprime Octagonal Number (3 x 7), also the first nontrivial 21-gonal Number, 21gon(2)## 22:

a Pentagonal Number P(4), also the Hexagonal Pyramidal Number HPyr(3), also the first nontrivial 22-gonal Number, 22gon(2)## 23:

the first nontrivial 23-gonal Number, 23gon(2)## 24:

Nonagonal Number N(3), also a Constructable Number because (2^3)x3 (power of two times product of Fermat Primes), also the first nontrivial 24-gonal Number, 24gon(2)## 25:

Square Number S(5), a Centered Square Number CS(4) also the first nontrivial 25-gonal number, 25gon(2)## 26:

the Heptagonal Pyramidal Number HepPyr(3), also the first nontrivial 26-gonal number, 26gon(2)## 27:

the Cube C(3), also the Decagonal Number D(3), also the first nontrivial 27-gonal Number, 27gon(2)## 28:

Triangular Number T(7), also Hexagonal Number H(4) also the first nontrivial 28-gonal Number, 28gon(2)## 29:

the first nontrivial 29-gonal Number, 29gon(2)## 30:

Hendecagonal Number, or 11-gonal number, Hen(3), also the Square Pyramidal Number SPyr(4), also a Constructable Number because 2x3x5 (power of two times product of Fermat Primes) also the first nontrivial 30-gonal Number, 30gon(2)## 31:

a Centered Cube Number CC(4), also Centered Triangle CT(5) also the Centered Pentagonal Number CP(4), also the first nontrivial 31-gonal Number, 31gon(2)## 32:

the first nontrivial Fifth Power, 2^5; also the first nontrivial 32-gonal Number, 32gon(2)## 33:

the 33rd Triangular Number is also a Dodecagonal Carmichael Number, also the first nontrivial 33-gonal Number, 33gon(2)## 34:

the Heptagonal Number Hep(4), also the first Semiprime Heptagonal Number (2 x 17), also the first nontrivial 34-gonal Number, 34gon(2); also a product of a power of two and a Fermat Prime, 2x17, and hence a regular 34-gon can be constructed by compass and straightedge, as Gauss first discovered (a Constructable Number)## 35:

a Pentagonal Number P(5), hence the first nontrivial Pentagonal Jonathan Number P(P(2)); also the Tetrahedral Number Tet(5), also the Pentatope Number Ptop(4); also the first nontrivial 35-gonal Number, 35gon(2)## 36:

Square Number S(6), also Triangular Number T(8), also 13-gonal Number 13gon(3), also the first nontrivial 36-gonal Number, 36gon(2)## 37:

Centered Hexagonal Number CH(4), also the first nontrivial 37-gonal Number, 37gon(2)## 38:

the first nontrivial Truncated Octahedral Number TruOct(2), also the first nontrivial 38-gonal Number, 38gon(2)## 39:

the 14-gonal Number, 14gon(3); also the first nontrivial 39-gonal Number, 39gon(2)## 40:

Octagonal Number O(4), also the Pentagonal Pyramidal Number PPyr(4); also the first nontrivial 40-gonal Number, 40gon(2) also a Constructable Number because (2^3)x5 (power of two times product of Fermat Primes)## 41:

a Centered Square Number CS(5), also the first nontrivial 41-gonal Number, 41gon(2)## 42:

the 15-gonal Number 15gon(3), also the first nontrivial 42-gonal Number, 42gon(2) [also see "A Hitchhiker's Guide to the Galaxy" by Douglas Adams]## 43:

The first "Uninteresting Number" -- isn't that interesting? This is sort of a joke among Mathematicians, as it proves that NO integers are uninteresting. {to be done: find a Polygonal Number for 43 other than the 43-gon} also the first nontrivial 43-gonal Number, 43gon(2)## 44:

Octahedral Number Octh(4), also the first nontrivial 44-gonal Number, 44gon(2)## 45:

Triangular Number T(9), also Hexagonal Number H(5), also 16-gonal Number 16gon(3)## 46:

Nonagonal Number N(4), also Centered Triangle CT(6)## 47:

The second "Uninteresting Number" in figurate number context## 48:

the 17-gonal Number 17gon(3), also a Constructable Number because (2^4)x3 (power of two times product of Fermat Primes)## 49:

a Square, S(7)## 50:

the Hexagonal Pyramidal Number HPyr(4)## 51:

the Pentagonal Number P(6), also the Centered Pentagonal Number CP(5), also the 18-gonal Number 18gon(3), also the Stella Octangula Number StOct(3), also a Constructable Number because a product of Fermat Primes, 3x17,## 52:

the Decagonal Number D(4)## 53:

The third "Uninteresting Number" in figurate number context## 54:

19-gonal Number 19gon(3)## 55:

the square root of a Square Dodecagonal Number, also the Square Pyramidal Number SPyr(5), also the first nontrival Square Pyramidal Jonathan Number SPyr(Spyr(2)), also Triangular Number T(10), also the 5th Semiprime Triangular Number, also Heptagonal Number Hep(5), also the 2nd Semiprime Heptagonal Number (5 x 11)## 56:

the Tetrahedral Number Tet(6)## 57:

the 20-gonal Number 20gon(3)## 58:

Hendecagonal Number, or 11-gonal number, Hen(4)## 59:

## 60:

21-gonal Number 21gon(3), also the Heptagonal Pyramidal Number HepPyr(4), also a Constructable Number because (2^2)x3x5 (power of two times product of Fermat Primes)## 61:

a Centered Square Number CS(6), also Centered Hexagonal Number CH(5)## 63:

the 22-gonal Number 22gon(3)## 64:

a Square Dodecagonal Number, S(8) and Dod(?), also Centered Triangle CT(7)## 65:

Octagonal Number O(5), also the 2nd Semiprime Octagonal Number (5 x 13), also Rhombic Dodecahedral Number RhoDod(3)## 66:

the Triangular Number T(11), also the 23-gonal Number 23gon(3)## 68:

the Truncated Tetrahedral Number TruTet(3), also a Constructable Number because (2^2)x17 (power of two times product of Fermat Primes)## 69:

the 24-gonal Number 24gon(3)## 70:

the Pentatope Number Ptop(5), also the first nontrivial Pentatope Jonathan Number Ptop(Ptop(2)), also Pentagonal Number P(7), also 13-gonal Number 13gon(4)## 72:

the 25-gonal number, 25gon(3)## 75:

Nonagonal Number N(5), also the 26-gonal Number, 26gon(3), also the Pentagonal Pyramidal Number PPyr(5)## 76:

the Centered Pentagonal Number CP(6), also the 14-gonal Number, 14gon(4)## 78:

the Triangular Number T(12), also the 27-gonal Number, 27gon(3)## 80:

a Constructable Number because (2^4)x5 (power of two times product of Fermat Primes)## 81:

Biquadratic Number (4th power) 3^4, because a Square, S(9), of a Square also Heptagonal Number Hep(5), also the 28-gonal Number, 28gon(3)## 82:

the 15-gonal Number 15gon(4)## 84:

the Tetrahedral Number Tet(7), also the 29-gonal Number 29gon(3)## 85:

Octahedral Number Octh(5) also a Centered Square Number CS(7), also Centered Triangle CT(8), also Decagonal Number D(5), also the 2nd Semiprime Decagonal Number (5 x 17), {to be done: need to insert Decagonal numbers in this series}, also a Constructable Number because 5x17, a power of two times a product of Fermat Primes## 87:

30-gonal Number 30gon(3)## 88:

the 16-gonal Number 16gon(4)## 90:

the 31-gonal Number 31gon(3)## 91:

the Triangular Number T(13), also the 6th Semiprime Triangular Number, also the Hexagonal Number H(7), also the 3rd Semiprime Hexagonal Number, also a Centered Cube Number CC(4), also Centered Hexagonal Number CH(6), also the Square Pyramidal Number SPyr(6), the 91st Dodecagonal Number, Dod(91) is also a Triangular Carmichael Number## 92:

the Pentagonal Number P(8)## 93:

the 32-gonal Number 32gon(3)## 94:

the 17-gonal Number 17gon(4)## 95:

Hendecagonal Number, or 11-gonal number, Hen(5); also the Hexagonal Pyramidal Number HPyr(5)## 96:

the Octagonal Number O(6), also the 33-gonal Number 33gon(3), also a Constructable Number because (2^5)x3 (power of two times product of Fermat Primes)## 99:

the 34-gonal Number 34gon(3)## 100:

a Square, S(10), also the 18-gonal Number 18gon(4)## 102:

the 35-gonal Number 35gon(3), also a Constructable Number because 2x3x17, a power of two times a product of Fermat Primes## 105:

a Triangular Dodecagonal Number, the Triangular Number T(14), the Dodecagonal Number D(__), also the 36-gonal Number 36gon(3)## 106:

the Centered Pentagonal Number CP(7), also 19-gonal Number 19gon(4)## 108:

the 37-gonal Number 37gon(3)## 109:

the Centered Triangle Number CT(9)## 111:

the Nonagonal Number N(6), also the 2nd Semiprime Nonagonal Number (3 x 37), also the 38-gonal Number 38gon(3)## 112:

Heptagonal Number Hep(7), also the first nontrivial Heptagonal Jonathan Number Hep(Hep(2)); also the 20-gonal Number 20gon(4)## 113:

the Centered Square Number CS(8)## 114:

the 39-gonal Number 39gon(3)## 115:

the Heptagonal Pyramidal Number HepPyr(5), also the 13-gonal Number 13gon(5)## 117:

the Pentagonal Number P(9); also the 40-gonal Number 40gon(3)## 118:

21-gonal Number 21gon(4)## 120:

5!, the Triangular Number T(15), also the Triangular Jonathan Number T(T(5)) also the Tetrahedral Number Tet(8), also the 41-gonal Number 41gon(3), also a Constructable Number because (2^3)x3x5 (power of two times product of Fermat Primes)## 121:

a Square, S(11)## 123:

the 42-gonal Number 42gon(3)## 124:

the Stella Octangula Number StOct(4), also the 22-gonal Number 22gon(4)## 125:

the Cube C(5), also the 14-gonal Number, 14gon(5)## 126:

the Pentagonal Pyramidal Number PPyr(6), also the Pentagonal Pyramidal Jonathan Number PPyr(PPyr(2)); also the Pentatope Number Ptop(6), also the Decagonal Number D(6), also the 43-gonal Number, 43gon(3)## 127:

the Centered Hexagonal Number CH(7), also a Centered Hexagonal Jonathan Number CH(CH(2))## 129:

the 44-gonal Number, 44gon(3)## 130:

the 23-gonal Number 23gon(4)## 133:

the Octagonal Number O(7), also the 3rd Semiprime Octagonal Number (7 x 19)## 135:

the 15-gonal Number 15gon(5)## 136:

the Triangular Number T(16), hence the Heterogeneous Jonathan Number T(S(4)), also the Centered Triangle CT(10), also the 24-gonal Number 24gon(4), also a Constructable Number because (2^3)x17 (power of two times product of Fermat Primes)## 137:

The reciprocal of a dimensionless constant in Atomic Physics is approximately 137.036. By dimensionless, I mean that this is just a number, unlike a distance in inches, which would be a different number of centimeters. Many people have wondered why this constant is what it is, and not larger or smaller; the question what is special about 137 has fascinated scientists. The constant of roughly 1/137.036 is called the Fine Structure Constant, because some of the things it measures are ceratin details of the spectrum of light emitted by glowing gases. Also: 137 = 10001 / 73.## 140:

the Square Pyramidal Number SPyr(7)## 141:

the Centered Pentagonal Number CP(8), also Hendecagonal Number, or 11-gonal number, Hen(6)## 142:

the 25-gonal number, 25gon(4)## 144:

a Square, S(12)## 145:

the Pentagonal Number P(10), also a Centered Square Number CS(9), also the 16-gonal Number 16gon(5)## 146:

Octahedral Number Octh(6), also an Octahedral Jonathan Number Octh(Octh(2))## 148:

the Heptagonal Number Hep(8), also the 26-gonal Number, 26gon(4)## 152:

note that 1, 152, 11552 are the first 3 terms of Sloane's A035823, and that 1152 comes from repeating the 1st and 2nd digits of 152...## 153:

the Triangular Number T(17)## 154:

Nonagonal Number N(7), also the 27-gonal Number, 27gon(4)## 155:

the 17-gonal Number 17gon(5)## 160:

the 28-gonal Number, 28gon(4), also a Constructable Number because (2^5)x5 (power of two times product of Fermat Primes)## 161:

the Hexagonal Pyramidal Number HPyr(6)## 162:

See Center of Gravity With Integer Coordinates also comes up three times in 6-ary Lyndon words with given trace and subtrace## 164:

10^2 + 8^2 = 20002 [base 3] = 1124 [base 5]. "E.164 addresses are used in networking ATMs. 164 comes up fairly early in Sloane's A000203.## 165:

the Tetrahedral Number Tet(9), hence the Heterogeneous Jonathan Number Tet(S(3)); also the 18-gonal Number 18gon(5)## 166:

Centered Triangle CT(11), also the 29-gonal Number 29gon(4)## 167:

12 x 167 = 2004 [this year]. 167 comes up in the problem of integer square tilings, a tiling of a square with smaller squares of integer sides. For every n > 1 consider all square tilings of an n x n square, and define: f(n) = the largest possible size of the smallest square g(n) = the smallest number of squares h(n) = the smalles value of the largest multiplicity of any square needed. Them for prime P, f(167) = 21 (with the largest possible minimum square), and f(167) = 21 (with the fewest possible squares). See Erich Friedman's Mathmagic## 169:

a Square S(13), also the 169th Dodecagonal Number is also a Square, also Centered Hexagonal Number CH(8)## 170:

a Constructable Number because 2x5x17, a power of two times a product of Fermat Primes## 171:

the Triangular Number T(18), also the 13-gonal Number 13gon(6)## 172:

30-gonal Number 30gon(4)## 175:

Rhombic Dodecahedral Number RhoDod(4), also the Decagonal Number D(7), also the 19-gonal Number 19gon(5)## 176:

the Pentagonal Number P(11), also Octagonal Number O(8), also Octagonal Jonathan Number O(O(2))## 173:

comes up twice in Erich Friedman's Mathmagic## 178:

the 31-gonal Number 31gon(4)## 179:

comes up in the theory of Odd Greedy Expanisons {to be done}## 180:

the Truncated Tetrahedral Number TruTet(4)## 181:

the Centered Pentagonal Number CP(9), also a Centered Square Number CS(10)## 184:

the 32-gonal Number 32gon(4)## 185:

the 20-gonal Number 20gon(5); "every integer exceeding 185 is representable [given in the proof of proposition 4] "d-complete sequences of integers", Erdos & Lewin, Mathematics of Computation, Vol.65, No.214, Apr 1996, pp.837-840.## 186:

the 14-gonal Number, 14gon(6)## 189:

a Centered Cube Number CC(5), also the Heptagonal Number Hep(9)## 190:

the Triangular Number T(19), also the 33-gonal Number 33gon(4)## 192:

a Constructable Number because (2^6)x3 (power of two times product of Fermat Primes)## 195:

21-gonal Number 21gon(5)## 196:

a Square Hendecagonal Number, Hen(7)=S(14) Hendecagonal Number, or 11-gonal number, Hen(7); also the 34-gonal Number 34gon(4), also the Heptagonal Pyramidal Number HepPyr(6); also the Pentagonal Pyramidal Number PPyr(7)## 199:

Centered Triangle CT(12)## 201:

the Truncated Octahedral Number TruOct(3), also the 15-gonal Number 15gon(6)## 202:

the 35-gonal Number 35gon(4)## 204:

the square root of a square triangular number (see Sloan A001110), also the Square Pyramidal Number SPyr(8), also Nonagonal Number N(8), {to be done: insert additional numbers in this sequence}; also a Constructable Number because (2^2)x3x17, a power of two times a product of Fermat Primes## 205:

the 22-gonal Number 22gon(5)## 208:

the 36-gonal Number 36gon(4)## 210:

a Triangular Pentagonal Number, also the Pentagonal Number P(12), also the Pentagonal Jonathan Number P(P(3)), also the Pentatope Number Ptop(7), also the Triangular Number T(20)## 214:

the 37-gonal Number 37gon(4)## 215:

the 23-gonal Number 23gon(5)## 216:

the Cube C(6), also the 16-gonal Number 16gon(6)## 217:

Centered Hexagonal Number CH(9)## 220:

the Tetrahedral Number Tet(10), also the Tetrahedral Jonathan Number Tet(Tet(3)), also the 38-gonal Number 38gon(4), also the Heterogeneous Jonathan Number 38gon(S(2))## 221:

a Centered Square Number CS(11) also 1!^2 + 2!^2 + 3!^2## 225:

a Square, S(15), also Octagonal Number O(9), also the Heterogeneous Jonathan Number O(S(3)), also the 24-gonal Number 24gon(5)## 226:

the Centered Pentagonal Number CP(10), also the 39-gonal Number 39gon(4)## 231:

Octahedral Number Octh(7), the Triangular Number T(21), also the Triangular Jonathan Number T(2,6) = T(T(6)), also the Triangular Jonathan Number T(3,3) = T(T(T(3))), also the Triangular Jonathan Number T(4,2) = T(T(T(T(2)))); also the 17-gonal Number 17gon(6)## 232:

the Decagonal Number D(8), also the Heterogeneous Jonathan Number D(C(2)); also the 40-gonal Number 40gon(4), also the Heterogeneous Jonathan Number 40gon(S(2))## 235:

Centered Triangle CT(13), also the Heptagonal Number Hep(10), also the 3rd Semiprime Heptagonal Number (5 x 47), also the 25-gonal number, 25gon(5)## 238:

the 13-gonal Number 13gon(7), also the 41-gonal Number 41gon(4)## 240:

a Constructable Number because (2^4)x3x5 (power of two times product of Fermat Primes)## 244:

the 42-gonal Number 42gon(4)## 245:

the Stella Octangula Number StOct(5), also the 26-gonal Number, 26gon(5)## 246:

the 18-gonal Number 18gon(6)## 247:

the Pentagonal Number P(13)## 250:

the 43-gonal Number, 43gon(4), also the Heterogeneous Jonathan Number 43gon(S(2))## 252:

the Hexagonal Pyramidal Number HPyr(7), also the Hexagonal Pyramidal Jonathan Number HPyr(HPyr(2))## 253:

the Triangular Number T(22)## 255:

the 27-gonal Number, 27gon(5), also a Constructable Number because 3x5x17, a power of two (in this case 2^0=1) times a product of Fermat Primes## 256:

the first nontrivial 8th power, 2^8 hence Biquadratic Number (4th power) 4^4 hence a Square, S(16); also the 44-gonal Number, 44gon(4), also the Heterogeneous Jonathan Number 44gon(S(2))## 257:

a Constructable Number because 257 (a Fermat Prime) is a power of two times a product of Fermat Primes## 259:

the 14-gonal Number 14gon(7)## 260:

Hendecagonal Number, or 11-gonal number, Hen(8)## 261:

Nonagonal Number N(9), also the Nonagonal Jonathan Number N(N(2)); also the 19-gonal Number 19gon(6)## 265:

a Centered Square Number CS(12), also the 28-gonal Number, 28gon(5)## 271:

Centered Hexagonal Number CH(10)## 272:

a Constructable Number because (2^4)x17 (power of two times product of Fermat Primes)## 274:

Centered Triangle CT(14)## 275:

the 29-gonal Number 29gon(5)## 276:

the Centered Pentagonal Number CP(11), also the Triangular Number T(23), also the 20-gonal Number 20gon(6)## 280:

Octagonal Number O(10), also the 15-gonal Number 15gon(7)## 285:

the Square Pyramidal Number SPyr(9), also 30-gonal Number 30gon(5)## 286:

the Tetrahedral Number Tet(11) also the 286th Triangular Number is also a Dodecagonal Carmichael Number; also the Heptagonal Number Hep(11)## 287:

the Pentagonal Number P(14), also the first Semiprime Pentagonal Number (7 x 41)## 288:

the 288th triangular number is also a square (see Sloan A001110); also the Pentagonal Pyramidal Number PPyr(8); Also see "Heron Tetrahedra." Other numbers which come up in Heron simplices include these combinations of edge lengths: (203,195,148) and (888,875,533).## 289:

a Square, S(17)## 291:

21-gonal Number 21gon(6)## 295:

the 31-gonal Number 31gon(5)## 297:

the Decagonal Number D(9), also the Heterogeneous Jonathan Number D(S(3))## 300:

the Triangular Number T(24)## 301:

the 16-gonal Number 16gon(7) also the first Odd Roman Numeral, alphabetically. See Numbers in Recreational Linguistics## 305:

the 32-gonal Number 32gon(5)## 306:

the 22-gonal Number 22gon(6)## 308:

the Heptagonal Pyramidal Number HepPyr(7)## 313:

a Centered Square Number CS(13)## 315:

the 33-gonal Number 33gon(5)## 316:

Centered Triangle CT(15), also the 13-gonal Number 13gon(8), also the Heterogeneous Jonathan Number 13gon(C(2))## 320:

a Constructable Number because (2^6)x5 (power of two times product of Fermat Primes)## 321:

the 23-gonal Number 23gon(6)## 322:

the 17-gonal Number 17gon(7)## 324:

a Square, S(18)## 325:

the Triangular Number T(25), hence the Heterogeneous Jonathan Number T(S(5)), also the Nonagonal Number N(10), also the 34-gonal Number 34gon(5)## 330:

the Pentatope Number Ptop(8), also the Pentagonal Number P(15)## 331:

the Centered Pentagonal Number CP(12), also Centered Hexagonal Number CH(11)## 333:

Hendecagonal Number, or 11-gonal number, Hen(9)## 335:

the 35-gonal Number 35gon(5)## 336:

the 24-gonal Number 24gon(6)## 340:

a Constructable Number because (2^2)x5x17, a power of two times a product of Fermat Primes## 341:

a Centered Cube Number CC(6), also Octagonal Number O(11), also the 4th Semiprime Octagonal Number (11 x 31)## 342:

the Heptagonal Number Hep(12)## 343:

the Cube C(7), also the 18-gonal Number 18gon(7)## 344:

Octahedral Number Octh(8), also the 14-gonal Number 14gon(8)## 345:

the 36-gonal Number 36gon(5)## 351:

the Triangular Number T(26), also the 25-gonal number, 25gon(6)## 355:

the 37-gonal Number 37gon(5)## 361:

Centered Triangle CT(16), also a Square, S(19)## 364:

the Tetrahedral Number Tet(12), also the 19-gonal Number 19gon(7)## 365:

a Centered Square Number CS(14), also the 38-gonal Number 38gon(5)## 366:

the 26-gonal Number, 26gon(6)## 369:

Rhombic Dodecahedral Number RhoDod(5)## 370:

the Decagonal Number D(10)## 372:

the Hexagonal Pyramidal Number HPyr(8), also the 15-gonal Number 15gon(8), also the Heterogeneous Jonathan Number 15gon(C(2))## 375:

the Truncated Tetrahedral Number TruTet(5), also the 39-gonal Number 39gon(5)## 376:

the Pentagonal Number P(16)## 377:

the square root of a Square Dodecagonal Number## 378:

the Triangular Number T(27), hence the Heterogeneous Triangular Number T(C(3))## 381:

the 27-gonal Number, 27gon(6)## 384:

a Constructable Number because (2^7)x3 (power of two times product of Fermat Primes)## 385:

the Square Pyramidal Number SPyr(10), also the 20-gonal Number 20gon(7), also the 40-gonal Number 40gon(5)## 391:

the Centered Pentagonal Number CP(13)## 395:

the 41-gonal Number 41gon(5)## 396:

the Nonagonal Number N(11), also the 28-gonal Number, 28gon(6)## 397:

Centered Hexagonal Number CH(12)## 400:

a Square, S(20), also the 16-gonal Number 16gon(8), also Heterogeneous Jonathan Number 16gon(C(2))## 403:

the Heptagonal Number Hep(13), also the 4th Semiprime Heptagonal Number (13 x 51)## 405:

the Pentagonal Pyramidal Number PPyr(9), also the 13-gonal Number 13gon(9), also the Heterogeneous Jonathan Number 13gon(S(3)); also the 42-gonal Number 42gon(5)## 406:

the Triangular Number T(28), hence the Triangular Jonathan Number T(T(7)); also 21-gonal Number 21gon(7)## 408:

a Constructable Number because (2^3)x3x17, a power of two times a product of Fermat Primes, also Octagonal Number O(12)## 409:

Centered Triangle CT(17)## 411:

the 29-gonal Number 29gon(6)## 415:

Hendecagonal Number, or 11-gonal number, Hen(10); also the 43-gonal Number, 43gon(5)## 421:

a Centered Square Number CS(15)## 425:

the Pentagonal Number P(17), also the 44-gonal Number, 44gon(5)## 426:

the Stella Octangula Number StOct(6), also 30-gonal Number 30gon(6)## 427:

the 22-gonal Number 22gon(7)## 428:

the 17-gonal Number 17gon(8), also the Heterogeneous Jonathan Number 17gon(C(2))## 435:

the Triangular Number T(29)## 441:

a Square, S(21), also the Heterogeneous Jonathan Number S(T(6)), also the Heterogeneous Jonathan Number S(T(T(3))) also the Heterogeneous Jonathan Number S(T(T(T(2)))) also the 14-gonal Number 14gon(9), also the Heterogeneous Jonathan Number 14gon(S(3)); also the 31-gonal Number 31gon(6)## 448:

the 23-gonal Number 23gon(7)## 451:

the Decagonal Number D(11), see Ray Bradbury's "Farenheit 451" also the 3rd Semiprime Decagonal Number (11 x 41)## 455:

the Tetrahedral Number Tet(13)## 456:

the Centered Pentagonal Number CP(14); also the 18-gonal Number 18gon(8), also the Heterogeneous Jonathan Number 18gon(C(2)); also the 32-gonal Number 32gon(6), also the Heptagonal Pyramidal Number HepPyr(8), also the Heptagonal Pyramidal Jonathan Number HepPyr(HepPyr(2))## 460:

Centered Triangle CT(18)## 465:

the Triangular Number T(30)## 469:

Centered Hexagonal Number CH(13), also the Heptagonal Number Hep(14), also the 5th Semiprime Heptagonal Number (7 x 67), also the 24-gonal Number 24gon(7)## 471:

the 33-gonal Number 33gon(6)## 474:

the Nonagonal Number N(12)## 477:

the Pentagonal Number P(18), also the 15-gonal Number 15gon(9), also the Heterogeneous Jonathan Number 15gon(S(3))## 480:

a Constructable Number because (2^5)x3x5 (power of two times product of Fermat Primes)## 481:

a Centered Square Number CS(16), also Octagonal Number O(13), also the 5th Semiprime Octagonal Number (13 x 37)## 484:

a Square, S(22), also the 19-gonal Number 19gon(8), also the Heterogeneous Jonathan Number 19gon(C(8))## 486:

the 34-gonal Number 34gon(6)## 489:

Octahedral Number Octh(9)## 490:

the 25-gonal number, 25gon(7)## 495:

the Pentatope Number Ptop(9)## 496:

the Triangular Number T(31)## 501:

the 35-gonal Number 35gon(6)## 505:

the 13-gonal Number 13gon(10)## 506:

Hendecagonal Number, or 11-gonal number, Hen(11), also the Square Pyramidal Number SPyr(11)## 510:

a Constructable Number because 2x3x5x17, a power of two times a product of Fermat Primes## 511:

the 26-gonal Number, 26gon(7)## 512:

the Cube C(8), since 8 is itself a cube, this is C(C(2)) = 2^9 the first nontrivial 9-dimensional Hypercube; also the 20-gonal Number 20gon(8), also the Heterogeneous Jonathan Number 20gon(C(2))## 513:

the 16-gonal Number 16gon(9), also the Heterogeneous Jonathan Number 16gon(S(3))## 514:

Centered Triangle CT(19), also a Constructable Number because 2x257, a power of two times a product of Fermat Primes## 516:

the 36-gonal Number 36gon(6)## 525:

the Hexagonal Pyramidal Number HPyr(9)## 526:

the Centered Pentagonal Number CP(15)## 528:

the Triangular Number T(32)## 529:

a Square, S(23)## 531:

the 37-gonal Number 37gon(6)## 532:

the Pentagonal Number P(19), also the 27-gonal Number, 27gon(7)## 540:

the Heptagonal Number Hep(15), also the Decagonal Number D(12), also the 21-gonal Number 21gon(8), also the Heterogeneous Jonathan Number 21gon(C(2))## 544:

a Constructable Number because (2^5)x17 (power of two times product of Fermat Primes)## 545:

the Centered Square Number CS(17)## 546:

the 38-gonal Number 38gon(6)## 547:

Centered Hexagonal Number CH(14)## 549:

the 17-gonal Number 17gon(9), also the Heterogeneous Jonathan Number 17gon(S(3))## 550:

the Pentagonal Pyramidal Number PPyr(10), also the 14-gonal Number 14gon(10)## 553:

the 28-gonal Number 28gon(7)## 559:

a Centered Cube Number CC(7), also the Nonagonal Number N(13), also the 3rd Semiprime Nonagonal Number (13 x 43)## 560:

the Tetrahedral Number Tet(14), also Octagonal Number O(14)## 561:

the smallest Carmichael Number; also a Triangular Number, and Dodecagonal Number the Triangular Number T(33); also the 39-gonal Number 39gon(6)## 568:

the 22-gonal Number 22gon(8), also the Heterogeneous Jonathan Number 22gon(C(2))## 571:

Centered Triangle CT(20)## 574:

the 29-gonal Number 29gon(7)## 576:

a Square, S(24); also the 40-gonal Number 40gon(6)## 585:

the 18-gonal Number 18gon(9), also the Heterogeneous Jonathan Number 18gon(S(3))## 586:

the Truncated Octahedral Number TruOct(4)## 588:

588^2 + 2353^2 = 5882353. "The Hermite-Serret Algorithm and 12^2 + 33^2 = 1233", Alf van der Poorten.## 590:

the Pentagonal Number P(20)## 591:

the 41-gonal Number 41gon(6)## 595:

the Triangular Number T(34), also the 15-gonal Number 15gon(10), also the 30-gonal Number 30gon(7)## 596:

the 23-gonal Number 23gon(8), also the Heterogeneous Jonathan Number 23gon(C(2))## 601:

the Centered Pentagonal Number CP(16)## 606:

Hendecagonal Number, or 11-gonal number, Hen(12); also the 42-gonal Number 42gon(6)## 613:

a Centered Square Number CS(18)## 616:

the Heptagonal Number Hep(16), note in base 10 how the same digits are used in the index and the Hep, also the Heterogeneous Jonathan Number Hep(S(4)); also the 13-gonal Number 13gon(11), also the 31-gonal Number 31gon(7)## 617:

1!^2 + 2!^2 + 3!^2 + 4!^2## 619:

1! - 2! + 3! - 4! + 5! - 6!## 621:

the 19-gonal Number 19gon(9), also the Heterogeneous Jonathan Number 19gon(S(3)); also the 43-gonal Number, 43gon(6)## 624:

the 24-gonal Number 24gon(8), also the Heterogeneous Jonathan Number 24gon(C(2))## 625:

the Square S(25) hence Biquadratic Number (4th power) 5^4 hence a Square, S(25)## 630:

the Triangular Number T(35)## 631:

Centered Triangle CT(21), also Centered Hexagonal Number CH(15)## 636:

the 44-gonal Number, 44gon(6), starts and ends with index and ends with square on index (base 10)## 637:

the Decagonal Number D(13), also the 32-gonal Number 32gon(7)## 640:

the 16-gonal Number 16gon(10), also a Constructable Number because (2^7)x5 (power of two times product of Fermat Primes)## 645:

the Heptagonal Pyramidal Number HepPyr(9), also Octagonal Number O(15)## 650:

the Square Pyramidal Number SPyr(12)## 651:

the Pentagonal Number P(21), also the Nonagonal Number N(14)## 652:

the 25-gonal number, 25gon(8)## 657:

the 20-gonal Number 20gon(9), also the Heterogeneous Jonathan Number 20gon(S(3))## 658:

the 33-gonal Number 33gon(7)## 666:

the Triangular Number T(36), hence the Triangular Jonathan Number T(T(8)), also the Heterogeneous Jonathan Number T(S(6)), also the Heterogeneous Jonathan Number T(T(C(2)))## 670:

Octahedral Number Octh(10)## 671:

Rhombic Dodecahedral Number RhoDod(6), also the 14-gonal Number 14gon(11)## 676:

a Square, S(26), also the Truncated Tetrahedral Number TruTet(6)## 679:

the Stella Octangula Number StOct(7), also the 34-gonal Number 34gon(7)## 680:

the Tetrahedral Number Tet(15), also the 26-gonal Number, 26gon(8), also a Constructable Number because (2^3)x5x17, a power of two times a product of Fermat Primes## 681:

the Centered Pentagonal Number CP(17)## 685:

a Centered Square Number CS(19), also the 17-gonal Number 17gon(10)## 693:

the 21-gonal Number 21gon(9), also the Heterogeneous Jonathan Number 21gon(S(3))## 694:

Centered Triangle CT(22)## 697:

the Heptagonal Number Hep(17), also the 6th Semiprime Heptagonal Number (17 x 41)## 700:

the 35-gonal Number 35gon(7)## 703:

the Triangular Number T(37), also the Hexagonal Number H(19), also the 4th Semiprime Hexagonal Number (19 x 37),## 708:

the 27-gonal Number, 27gon(8)## 715:

a Pentagonal Hendecagonal Number, P(22), also the Hendecagonal Number, or 11-gonal number, Hen(13); also the Hexagonal Pyramidal Number HPyr(10); also the Pentatope Number Ptop(10)## 721:

Centered Hexagonal Number CH(16), also the 36-gonal Number 36gon(7)## 726:

the Pentagonal Pyramidal Number PPyr(11), also the 15-gonal Number 15gon(11)## 729:

the Cube C(9), since 9 is itself a square, this is C(S(3)) = 3^6 a sixth power, hence a Square, S(27); also the 22-gonal Number 22gon(9), also the Heterogeneous Jonathan Number 22gon(S(3))## 730:

the 18-gonal Number 18gon(10)## 736:

Octagonal Number O(16), also the Heterogeneous Jonathan Number O(S(4)), also the 28-gonal Number 28gon(8)## 738:

the 13-gonal Number 13gon(12)## 741:

the Triangular Number T(38)## 742:

the Decagonal Number D(14), also the 37-gonal Number 37gon(7)## 750:

the Nonagonal Number N(15)## 760:

Centered Triangle CT(23)## 761:

a Centered Square Number CS(20)## 763:

the 38-gonal Number 38gon(7)## 764:

the 29-gonal Number 29gon(8), also the Heterogeneous Jonathan Number 29gon(C(2))## 765:

the 23-gonal Number 23gon(9), also the Heterogeneous Jonathan Number 23gon(S(3))## 766:

the Centered Pentagonal Number CP(18)## 768:

a Constructable Number because (2^8)x3 (power of two times product of Fermat Primes)## 771:

a Constructable Number because 3x257, a power of two (here 2^0=1) times a product of Fermat Primes## 775:

the 19-gonal Number 19gon(10); the 775th Dodecagonal Number is also a Triangular Number## 780:

the Triangular Number T(39)## 781:

the 16-gonal Number 16gon(11)## 782:

a Pentagonal Number P(23)## 783:

the Heptagonal Number Hep(18), also the Heptagonal Jonathan Number Hep(Hep(3))## 784:

a Square, S(28), also the Heterogeneous Jonathan Number S(T(7)); also the 39-gonal Number 39gon(7)## 792:

the 30-gonal Number 30gon(8), also the Heterogeneous Jonathan Number 30gon(C(2))## 801:

the 24-gonal Number 24gon(9), also the Heterogeneous Jonathan Number 24gon(S(3))## 804:

the 14-gonal Number 14gon(12)## 805:

the 40-gonal Number 40gon(7)## 809:

1!^5 + 2!^4 + 3!^3 + 4!^2 + 5!^1## 816:

the Tetrahedral Number Tet(16), also a Constructable Number because (2^4)x3x17, a power of two times a product of Fermat Primes## 817:

Centered Hexagonal Number CH(17)## 819:

the Square Pyramidal Number SPyr(13)## 820:

the Triangular Number T(40), also the 20-gonal Number 20gon(10), also the 31-gonal Number 31gon(8), also the heterogeneous Jonathan Number 31gon(C(2))## 826:

the 41-gonal Number 41gon(7)## 829:

Centered Triangle CT(24)## 833:

Hendecagonal Number, or 11-gonal number, Hen(14), also Octagonal Number O(17)## 836:

the 17-gonal Number 17gon(11)## 837:

the 25-gonal number, 25gon(9), also the Heterogeneous Jonathan Number 25gon(S(3))## 841:

a Centered Square Number CS(21), also a Square, S(29)## 847:

the 42-gonal Number 42gon(7)## 848:

the 32-gonal Number 32gon(8), also the Heterogeneous Jonathan Number 32gon(C(2))## 852:

a Pentagonal Number P(24)## 855:

a Centered Cube Number CC(8), also the Decagonal Number D(15)## 856:

the Centered Pentagonal Number CP(19), also the Nonagonal Number N(16), also the Heterogeneous Jonathan Number N(S(4))## 861:

the Triangular Number T(41)## 865:

the 21-gonal Number 21gon(10)## 868:

the 43-gonal Number, 43gon(7)## 870:

the 15-gonal Number 15gon(12)## 871:

the 13-gonal Number 13gon(13)## 873:

the 26-gonal Number, 26gon(9)## 874:

the Heptagonal Number Hep(19)## 876:

the 33-gonal Number 33gon(8), also the Heterogeneous Jonathan Number 33gon(C(2))## 880:

the Heptagonal Pyramidal Number HepPyr(10)## 889:

the 44-gonal Number, 44gon(7)## 891:

Octahedral Number Octh(11), also the 18-gonal Number 18gon(11)## 900:

a Square, S(30)## 901:

Centered Triangle CT(25)## 903:

the Triangular Number T(42)## 904:

the 34-gonal Number 34gon(8), also the Heterogeneous Jonathan Number 34gon(C(2))## 909:

the 27-gonal Number, 27gon(9), also the Heterogeneous Jonathan Number 27gon(S(3))## 910:

the 22-gonal Number 22gon(10)## 919:

Centered Hexagonal Number CH(18), also a Palindrome## 925:

a Pentagonal Number P(25), also the Heterogeneous Jonathan Number P(S(5)), also a Centered Square Number CS(22)## 932:

the 35-gonal Number 35gon(8), also the Heterogeneous Jonathan Number 35gon(C(2))## 936:

the Pentagonal Pyramidal Number PPyr(12), also Octagonal Number O(18), also the 16-gonal Number 16gon(12)## 945:

the 28-gonal Number 28gon(9)## 946:

the Hexagonal Pyramidal Number HPyr(11), also the Triangular Number T(43), also the 19-gonal Number 19gon(11)## 949:

the 14-gonal Number 14gon(13)## 951:

the Centered Pentagonal Number CP(20)## 955:

the 23-gonal Number 23gon(10)## 960:

Hendecagonal Number, or 11-gonal number, Hen(15), also the 36-gonal Number 36gon(8), also the Heterogeneous Jonathan Number 36gon(C(2)), also a Constructable Number because (2^6)x3x5 (power of two times product of Fermat Primes)## 961:

a Square, S(31)## 969:

the Tetrahedral Number Tet(17), also the Nonagonal Number N(17)## 970:

the Heptagonal Number Hep(20)## 976:

Centered Triangle CT(26), also the Decagonal Number D(16), also the Heterogeneous Jonathan Number D(S(4))## 981:

the 29-gonal Number 29gon(9), also the Heterogeneous Jonathan Number 29gon(S(3))## 988:

the 37-gonal Number 37gon(8), also the Heterogeneous Jonathan Number 37gon(C(2))## 990:

the Triangular Number T(44)## 1000:

the Cube C(10), also the 24-gonal Number 24gon(10)## 1001:

a Pentagonal Number P(26), also the 20-gonal Number 20gon(11), also the same reversed or upside-down; also the Pentatope Number Ptop(11) also the product of of primes 7x11x13## 1002:

the 17-gonal Number 17gon(12)## 1005:

Rhombic Dodecahedral Number RhoDod(7), also, "One Thousand Five" is the smallest number whose English name uses the five vowels a, e, i, o, u, in any order. See Numbers in Recreational Linguistics## 1013:

a Centered Square Number CS(23)## 1015:

the Square Pyramidal Number SPyr(14), also the 13-gonal Number 13gon(14)## 1016:

the Stella Octangula Number StOct(8), also the 38-gonal Number 38gon(8), also the Heterogeneous Jonathan Number 38gon(C(2))## 1017:

the 30-gonal Number 30gon(9), also the Heterogeneous Jonathan Number 30gon(S(3))## 1020:

a Constructable Number because (2^2)x3x5x17, a power of two times a product of Fermat Primes## 1024:

a Square, S(32), but since 32 is a 5th power, this is a 10th power 2^10## 1025:

"One Thousand Twenty-Five" is the smallest number whose English name uses the six vowels a, e, i, o, u, y in any order. See Numbers in Recreational Linguistics## 1027:

Centered Hexagonal Number CH(19), also a Centered Hexagonal Jonathan Number CH(CH(3)), also the 15-gonal Number 15gon(13)## 1028:

a Constructable Number because (2^2)x257, a power of two times a product of Fermat Primes## 1035:

the Triangular Number T(45), hence the Triangular Jonathan Number T(T(9)), hence the Heterogeneous Jonathan Number T(T(S(3)))## 1140:

the Tetrahedral Number Tet(18)## 1044:

the 39-gonal Number 39gon(8), also the Heterogeneous Jonathan Number 39gon(C(2))## 1045:

Octagonal Number O(19), also the 25-gonal number, 25gon(10)## 1051:

the Centered Pentagonal Number CP(21)## 1053:

the 31-gonal Number 31gon(9), also the Heterogeneous Jonathan Number 31gon(S(3))## 1054:

the Centered Triangle CT(27)## 1056:

the 21-gonal Number 21gon(11)## 1068:

the 18-gonal Number 18gon(12)## 1071:

the Heptagonal Number Hep(21)## 1072:

the 40-gonal Number 40gon(8), also the Heterogeneous Jonathan Number 40gon(C(2))## 1080:

a Pentagonal Number P(27), also the Heterogeneous Jonathan Number P(C(3)), also the smallest number with 18 divisors## 1081:

the Triangular Number T(46)## 1084:

"One Thousand Eighty-Four" is the smallest number whose English name uses the five vowels a, e, i, o, u, in that order. See Numbers in Recreational Linguistics## 1088:

a Constructable Number because (2^6)x17 (power of two times product of Fermat Primes)## 1089:

a Square, S(33), also the Nonagonal Number N(18), also the 32-gonal Number 32gon(9), also the Heterogeneous Jonathan Number 32gon(S(3))## 1090:

the 26-gonal Number, 26gon(10)## 1096:

Hendecagonal Number, or 11-gonal number, Hen(16), hence the Heterogeneous Jonathan Number Hen(S(4))## 1100:

the 41-gonal Number 41gon(8), also the Heterogeneous Jonathan Number 41gon(C(2));## 1104:

One Thousand One Hundred Four is the smallest number whose English name is spelled with 25 letters. See Numbers in Recreational Linguistics## 1105:

a Centered Square Number CS(24), also the Decagonal Number D(17), also the 16-gonal Number 16gon(13)## 1106:

the Truncated Tetrahedral Number TruTet(7), also the 14-gonal Number 14gon(14), also the 14-gonal Jonathan Number 14gon(14gon(2))## 1111:

the 22-gonal Number 22gon(11), repunit with repunit index (base 10)## 1117:

One Thousand One Hundred Seventeen is the smallest number whose English name is spelled with 30 letters. See Numbers in Recreational Linguistics## 1125:

the 33-gonal Number 33gon(9), also the Heterogeneous Jonathan Number 33gon(S(3));## 1128:

the Triangular Number T(47), also the 42-gonal Number 42gon(8), also the Heterogeneous Jonathan Number 42gon(C(2))## 1134:

the 19-gonal Number 19gon(12)## 1135:

Centered Triangle CT(28), also the 27-gonal Number, 27gon(10)## 1141:

Centered Hexagonal Number CH(20)## 1156:

Octahedral Number Octh(12), the 1156th Dodecagonal Number is also a Square, also the Centered Pentagonal Number CP(22), also a Square, S(34), also the 43-gonal Number, 43gon(8), also the Heterogeneous Jonathan Number 43gon(C(2))## 1160:

Octagonal Number O(20)## 1161:

the 34-gonal Number 34gon(9), also the Heterogeneous Jonathan Number 34gon(S(3))## 1162:

a Pentagonal Number P(28)## 1166:

the Heptagonal Pyramidal Number HepPyr(11), also the 23-gonal Number 23gon(11)## 1170:

the 13-gonal Number 13gon(15)## 1176:

the Triangular Number T(48)## 1177:

the Heptagonal Number Hep(22), also the 7th Semiprime Heptagonal Number (11 x 107)## 1180:

the 28-gonal Number 28gon(10)## 1183:

the Pentagonal Pyramidal Number PPyr(13), also the 17-gonal Number 17gon(13)## 1184:

the 44-gonal Number 44gon(8), also the Heterogeneous Jonathan Number 44gon(C(2))## 1189:

the square root of a Square Triangular number (see Sloan A001110)## 1197:

the 15-gonal Number 15gon(14), also the 35-gonal Number 35gon(9), also the Heterogeneous Jonathan Number 35gon(S(3))## 1200:

the 20-gonal Number 20gon(12), starts with same 2 digits as index (base 10)## 1201:

a Centered Square Number CS(25)## 1216:

the Nonagonal Number N(19)## 1219:

Centered Triangle CT(29)## 1221:

the 24-gonal Number 24gon(11), number and index both start and end with 1 (base 10)## 1222:

the Hexagonal Pyramidal Number HPyr(12)## 1225:

a Square, S(35), also the Triangular Number T(49), hence a Square Triangular Number, also the Heterogeneous Jonathan Number T(S(7)), also the 29-gonal Number 29gon(10)## 1233:

the 36-gonal Number 36gon(9), also the Heterogeneous Jonathan Number 36gon(S(3))## 1240:

the Square Pyramidal Number SPyr(15)## 1241:

a Centered Cube Number CC(9), also Hendecagonal Number, or 11-gonal number, Hen(17)## 1242:

the Decagonal Number D(18)## 1247:

a Pentagonal Number P(29), also the 2nd Semiprime Pentagonal Number (29 x 47)## 1261:

Centered Hexagonal Number CH(21), also the 18-gonal Number 18gon(13)## 1266:

the 21-gonal Number 21gon(12), also the Centered Pentagonal Number CP(23)## 1269:

the 37-gonal Number 37gon(9), also the Heterogeneous Jonathan Number 37gon(S(3))## 1270:

the 30-gonal Number 30gon(10)## 1275:

the Triangular Number T(50), also the 14-gonal Number 14gon(15)## 1276:

the 25-gonal number, 25gon(11)## 1280:

a Constructable Number because (2^8)x5 (power of two times product of Fermat Primes)## 1281:

Octagonal Number O(21), also Octagonal Jonathan Number O(O(3))## 1285:

a Constructable Number because 5x257, a power of two (here 2^0=1) times a product of Fermat Primes## 1288:

the Heptagonal Number Hep(23), also the 16-gonal Number 16gon(14)## 1289:

the Truncated Octahedral Number TruOct(5)## 1296:

a Square, S(36), but since 36 is a Square, this is a Biquadratic 6^4, also the Heterogeneous Jonathan Number S(T(8))## 1301:

a Centered Square Number CS(26)## 1305:

the 38-gonal Number 38gon(9), also the Heterogeneous Jonathan Number 38gon(S(3))## 1306:

Centered Triangle CT(30)## 1315:

the 31-gonal Number 31gon(10)## 1326:

the Triangular Number T(51)## 1330:

the Tetrahedral Number Tet(19)## 1331:

the Cube C(11), also the 26-gonal Number, 26gon(11)## 1332:

the 22-gonal Number 22gon(12)## 1335:

a Pentagonal Number P(30)## 1336:

the 13-gonal Number 13gon(16), also the Heterogeneous Jonathan Number 13gon(S(4))## 1339:

the 19-gonal Number 19gon(13), 2-digit index is same as first 2 digits of 19-gonal, next 2 triple that## 1341:

the 39-gonal Number 39gon(9), also the Heterogeneous Jonathan Number 39gon(S(3))## 1350:

the Nonagonal Number N(20)## 1360:

the 32-gonal Number 32gon(10), also a Constructable Number because (2^4)x5x17, a power of two times a product of Fermat Primes## 1365:

the Pentatope Number Ptop(12)## 1369:

a Square, S(37)## 1377:

the 40-gonal Number 40gon(9), also the Heterogeneous Jonathan Number 40gon(S(3))## 1378:

the Triangular Number T(52)## 1379:

the 17-gonal Number 17gon(14)## 1380:

the 15-gonal Number 15gon(15), also the 15gonal Jonathan Number 15gon(15gon(2))## 1381:

the Centered Pentagonal Number CP(24)## 1386:

the 27-gonal Number, 27gon(11)## 1387:

Centered Hexagonal Number CH(22), also the Decagonal Number D(19), also the 4th Semiprime Decagonal Number (19 x 73)## 1395:

Hendecagonal Number, or 11-gonal number, Hen(18)## 1396:

Centered Triangle CT(31)## 1398:

the 23-gonal Number 23gon(12)## 1404:

the Heptagonal Number Hep(24)## 1405:

a Centered Square Number CS(27), also the 33-gonal Number 33gon(10)## 1408:

Octagonal Number O(22)## 1413:

the 41-gonal Number 41gon(9), also the Heterogeneous Jonathan Number 41gon(S(3));## 1417:

the 20-gonal Number 20gon(13)## 1426:

a Pentagonal Number P(31)## 1431:

the Triangular Number T(53)## 1441:

the 28-gonal Number 28gon(11)## 1444:

a Square, S(38)## 1449:

the Stella Octangula Number StOct(9), also the 42-gonal Number 42gon(9), also the Heterogeneous Jonathan Number 42gon(S(3))## 1450:

the 34-gonal Number 34gon(10)## 1456:

the 14-gonal Number 14gon(16), also the Heterogeneous Jonathan Number 14gon(S(4))## 1464:

the 24-gonal Number 24gon(12)## 1469:

Octahedral Number Octh(13)## 1470:

the Pentagonal Pyramidal Number PPyr(14), also the 18-gonal Number 18gon(14)## 1485:

the Triangular Number T(54), also the 16-gonal Number 16gon(15), also the 43-gonal Number, 43gon(9), also the Heterogeneous Jonathan Number 43gon(S(3))## 1489:

Centered Triangle CT(32)## 1491:

the Nonagonal Number N(21)## 1495:

the 21-gonal Number 21gon(13), also the 35-gonal Number 35gon(10)## 1496:

the Square Pyramidal Number SPyr(16), also the 29-gonal Number 29gon(11)## 1501:

the Centered Pentagonal Number CP(25)## 1508:

the Heptagonal Pyramidal Number HepPyr(12)## 1513:

a Centered Square Number CS(28), also the 13-gonal Number 13gon(17)## 1519:

Centered Hexagonal Number CH(23)## 1520:

a Pentagonal Number P(32)## 1521:

a Square, S(39); also the 44-gonal Number 44gon(9), also the Heterogeneous Jonathan Number 44gon(S(3))## 1525:

the Heptagonal Number Hep(25)## 1530:

the 25-gonal number, 25gon(12)## 1536:

a Constructable Number because (2^9)x3 (power of two times product of Fermat Primes)## 1540:

a Triangular, Hexagonal, Decagonal, Hendecagonal Pyramidal Number; T(55) = H(??) = D(20), also the Tetrahedral Number Tet(20), also the Tetrahedral Jonathan Number Tet(2,4) = Tet(Tet(4)), also the Tetrahedral Jonathan Number Tet(3,2) = Tet(Tet(Tet(2))), also the Triangular Number T(55), hence the Triangular Jonathan Number T(2,10) = T(T(10)), hence the Triangular Jonathan Number T(3,4) = T(T(T(4))); also the 36-gonal Number 36gon(10)## 1541:

Octagonal Number O(23), also the 6th Semiprime Octagonal Number (23 x 67)## 1542:

a Constructable Number because 2x3x257, a power of two times a product of Fermat Primes## 1547:

the Hexagonal Pyramidal Number HPyr(13)## 1551:

the 30-gonal Number 30gon(11)## 1558:

Hendecagonal Number, or 11-gonal number, Hen(19)## 1561:

the 19-gonal Number 19gon(14)## 1573:

the 22-gonal Number 22gon(13)## 1576:

the 15-gonal Number 15gon(16), also the Heterogeneous Jonathan Number 15gon(S(4))## 1585:

Centered Triangle CT(33), also the 37-gonal Number 37gon(10)## 1590:

the 17-gonal Number 17gon(15)## 1596:

the Triangular Number T(56), also the 26-gonal Number, 26gon(12)## 1600:

a Square, S(40)## 1606:

the 31-gonal Number 31gon(11)## 1617:

a Pentagonal Number P(33)## 1625:

a Centered Square Number CS(29)## 1626:

the Centered Pentagonal Number CP(26)## 1630:

the 38-gonal Number 38gon(10)## 1632:

a Constructable Number because (2^5)x3x17, a power of two times a product of Fermat Primes## 1639:

the Nonagonal Number N(22), also the the 4th Semiprime Nonagonal Number (11 x 149)## 1649:

the 14-gonal Number 14gon(17)## 1651:

the Heptagonal Number Hep(26), also the 8th Semiprime Heptagonal Number (13 x 127), also the 23-gonal Number 23gon(13)## 1652:

the 20-gonal Number 20gon(14)## 1653:

the Triangular Number T(57)## 1657:

Centered Hexagonal Number CH(24)## 1661:

the 32-gonal Number 32gon(11), a Palindrome with Palindrome index (base 10)## 1662:

the 27-gonal Number, 27gon(12)## 1675:

the 39-gonal Number 39gon(10)## 1680:

Octagonal Number O(24)## 1681:

the 1681st Triangular number is also a Square S(41) (see Sloan A001110); and both the initial pair of digits (16) and the final pair of digits (81) are squares.## 1684:

Centered Triangle CT(34)## 1688:

the Truncated Tetrahedral Number TruTet(8)## 1695:

Rhombic Dodecahedral Number RhoDod(8), also the 18-gonal Number 18gon(15)## 1696:

the 16-gonal Number 16gon(16), also the 16-gonal Jonathan Number 16gon(16gon(2))## 1701:

the Decagonal Number D(21), also the 13-gonal Number 13gon(18)## 1711:

the Triangular Number T(58)## 1716:

the 33-gonal Number 33gon(11)## 1717:

a Pentagonal Number P(34), also the 3rd Semiprime Pentagonal Number (17 x 101)## 1720:

the 40-gonal Number 40gon(10)## 1728:

the Cube C(12), also the 28-gonal Number 28gon(12)## 1729:

the 24-gonal Number 24gon(13), also a Carmichael Number; also a Dodecagonal Number, D(__); also a Centered Cube Number CC(10)## 1730:

Hendecagonal Number, or 11-gonal number, Hen(20)## 1741:

a Centered Square Number CS(30)## 1743:

the 21-gonal Number 21gon(14)## 1756:

the Centered Pentagonal Number CP(27)## 1764:

a Square, S(42)## 1765:

the 41-gonal Number 41gon(10)## 1770:

the Triangular Number T(59)## 1771:

the Tetrahedral Number Tet(21), also a Palindrome; also the 34-gonal Number 34gon(11)## 1782:

the Heptagonal Number Hep(27)## 1785:

the Square Pyramidal Number SPyr(17), also the 15-gonal Number 15gon(17)## 1786:

Centered Triangle CT(35)## 1794:

the Nonagonal Number N(23), also the 29-gonal Number 29gon(12)## 1800:

the Pentagonal Pyramidal Number PPyr(15), also the 19-gonal Number 19gon(15)## 1801:

Centered Hexagonal Number CH(25)## 1807:

the 25-gonal number, 25gon(13)## 1810:

the 42-gonal Number 42gon(10)## 1816:

the 17-gonal Number 17gon(16), also the Heterogeneous Jonathan Number 17gon(S(4))## 1820:

a Pentagonal Number P(35), and 16C4; also the Pentagonal Jonathan Number P(P(5)) = P(P(P(2))) also the Pentatope Number Ptop(13)## 1825:

Octagonal Number O(25), also the Heterogeneous Jonathan Number O(S(5))## 1826:

the 35-gonal Number 35gon(11)## 1830:

the Triangular Number T(60)## 1834:

Octahedral Number Octh(14), also the 22-gonal Number 22gon(14)## 1849:

a Square, S(43)## 1854:

the 14-gonal Number 14gon(18)## 1855:

the 43-gonal Number, 43gon(10)## 1860:

the 30-gonal Number 30gon(12)## 1861:

a Centered Square Number CS(31)## 1870:

the Decagonal Number D(22)## 1881:

the 36-gonal Number 36gon(11), the same number upside-down and also backwards, as is its index (base 10)## 1885:

the 26-gonal Number, 26gon(13)## 1891:

the Centered Pentagonal Number CP(28), also Centered Triangle CT(36), also the Triangular Number T(61), also the 5th Semiprime Hexagonal Number (31 x 61)## 1900:

the 13-gonal Number 13gon(19), 13-gonal starts with same 2 digits as index (base 10); also the 44-gonal Number 44gon(10)## 1905:

the 20-gonal Number 20gon(15)## 1911:

Hendecagonal Number, or 11-gonal number, Hen(21), also the Heptagonal Pyramidal Number HepPyr(13)## 1918:

the Heptagonal Number Hep(28)## 1920:

a Constructable Number because (2^7)x3x5 (power of two times product of Fermat Primes)## 1921:

the 16-gonal Number 16gon(17)## 1925:

the Hexagonal Pyramidal Number HPyr(14), also the 23-gonal Number 23gon(14)## 1926:

a Pentagonal Number P(36), also the Heterogenous Jonathan Number P(S(6)); also the 31-gonal Number 31gon(12)## 1936:

a Square, S(44); also the 18-gonal Number 18gon(16), also the Heterogenous Jonathan Number 18gon(S(4)); also the 37-gonal Number 37gon(11)## 1951:

Centered Hexagonal Number CH(26)## 1953:

the Triangular Number T(62)## 1956:

the Nonagonal Number N(24), also the Nonagonal Jonathan Number N(N(3))## 1963:

the 27-gonal Number, 27gon(13)## 1976:

Octagonal Number O(26)## 1985:

a Centered Square Number CS(32)## 1990:

the Stella Octangula Number StOct(10)## 1991:

the 38-gonal Number 38gon(11), like the 2-digit index this 4-digit number is a palindrome (base 10)## 1992:

the 32-gonal Number 32gon(12)## 1999:

Centered Triangle CT(37)## 2007:

the 15-gonal Number 15gon(18)## 2010:

the 21-gonal Number 21gon(15)## 2016:

the Triangular Number T(63), also the 24-gonal Number 24gon(14)## 2024:

the Tetrahedral Number Tet(22)## 2025:

a Square, S(45), also the Heterogeneous Jonathan Number S(T(9))## 2031:

the Centered Pentagonal Number CP(29)## 2035:

a Pentagonal Number P(37), and 17C4## 2040:

a Constructable Number because (2^3)x3x5x17, a power of two times a product of Fermat Primes## 2041:

the 28-gonal Number 28gon(13)## 2046:

the 39-gonal Number 39gon(11)## 2047:

the Decagonal Number D(23), also the 5th Semiprime Decagonal Number (23 x 89)## 2056:

the 19-gonal Number 19gon(16), also the Heterogeneous Jonathan Number 19gon(S(4)); also a Constructable Number because (2^3)x257, a power of two times a product of Fermat Primes## 2057:

the 17-gonal Number 17gon(17), also the 17-gonal Jonathan Number 17gon(17gon(2))## 2058:

the 33-gonal Number 33gon(12)## 2059:

the Heptagonal Number Hep(29), also the 9th Semiprime Heptagonal Number (29 x 71)## 2071:

the 14-gonal Number 14gon(19)## 2080:

the Triangular Number T(64), hence the Heterogeneous Jonathan Number T(S(8))## 2101:

Hendecagonal Number, or 11-gonal number, Hen(22); also the 40-gonal Number 40gon(11)## 2107:

Centered Hexagonal Number CH(27), also the 25-gonal number, 25gon(14)## 2109:

the Square Pyramidal Number SPyr(18)## 2110:

Centered Triangle CT(38), also the 13-gonal Number 13gon(20)## 2113:

a Centered Square Number CS(33)## 2115:

the 22-gonal Number 22gon(15), last two digits equals index (base 10)## 2116:

a Square, S(46)## 2119:

the 29-gonal Number 29gon(13)## 2124:

the 34-gonal Number 34gon(12)## 2125:

the Nonagonal Number N(25), also the Heterogeneous Jonathan Number N(S(5))## 2133:

Octagonal Number O(27), also Heterogeneous Jonathan Number O(C(3))## 2145:

the Triangular Number T(65)## 2147:

a Pentagonal Number P(38), also the 4th Semiprime Pentagonal Number (19 x 113)## 2156:

the 41-gonal Number 41gon(11)## 2160:

the 16-gonal Number 16gon(18)## 2176:

the Centered Pentagonal Number CP(30), also the Pentagonal Pyramidal Number PPyr(16), also the 20-gonal Number 20gon(16), also the Heterogeneous Jonathan Number 20gon(S(4)); also a Constructable Number because (2^7)x17 (power of two times product of Fermat Primes)## 2190:

the 35-gonal Number 35gon(12)## 2193:

the 18-gonal Number 18gon(17)## 2197:

the Cube C(13), also the 30-gonal Number 30gon(13)## 2198:

the 26-gonal Number, 26gon(14)## 2205:

the Heptagonal Number Hep(30)## 2209:

a Square, S(47)## 2211:

the Triangular Number T(66), hence Triangular Jonathan Number T(T(11)); also the 42-gonal Number 42gon(11), first 2 digits double the index, last 2 digits equal the index (base 10)## 2220:

the 23-gonal Number 23gon(15)## 2224:

Centered Triangle CT(39)## 2232:

the Decagonal Number D(24)## 2242:

the 15-gonal Number 15gon(19)## 2245:

a Centered Square Number CS(34)## 2255:

Octahedral Number Octh(15)## 2256:

the 36-gonal Number 36gon(12)## 2262:

a Pentagonal Number P(39)## 2266:

the 43-gonal Number, 43gon(11)## 2269:

Centered Hexagonal Number CH(28)## 2275:

the 31-gonal Number 31gon(13)## 2278:

the Triangular Number T(67)## 2289:

the 27-gonal Number, 27gon(14)## 2296:

Octagonal Number O(28), also the 21-gonal Number 21gon(16), also the Heterogeneous Jonathan Number 21gon(S(4))## 2300:

Hendecagonal Number, or 11-gonal number, Hen(23); also the 14-gonal Number 14gon(20), also the Tetrahedral Number Tet(23)## 2301:

the Nonagonal Number N(26)## 2304:

a Square, S(48)## 2313:

the 17-gonal Number 17gon(18)## 2321:

the 44-gonal Number 44gon(11)## 2322:

the 37-gonal Number 37gon(12)## 2325:

the 24-gonal Number 24gon(15)## 2326:

the Centered Pentagonal Number CP(31)## 2329:

the 19-gonal Number 19gon(17)## 2331:

a Centered Cube Number CC(11), also the 13-gonal Number 13gon(21)## 2341:

Centered Triangle CT(40)## 2346:

the Triangular Number T(68)## 2353:

the 32-gonal Number 32gon(13)## 2356:

the Heptagonal Number Hep(31)## 2360:

the Hexagonal Pyramidal Number HPyr(15)## 2380:

the Heptagonal Pyramidal Number HepPyr(14), also the Pentatope Number Ptop(14), also the Pentagonal Number P(40), also the 28-gonal Number 28gon(14)## 2381:

a Centered Square Number CS(35)## 2388:

the 38-gonal Number 38gon(12)## 2401:

a Square, S(49), also the Biquadratic 7^4## 2406:

the Truncated Octahedral Number TruOct(6)## 2413:

the 16-gonal Number 16gon(19)## 2415:

the Triangular Number T(69)## 2416:

the 22-gonal Number 22gon(16), also the Heterogeneous Jonathan Number 22gon(S(4))## 2425:

the Decagonal Number D(25), also the Heterogeneous Jonathan Number D(S(5))## 2430:

the 25-gonal number, 25gon(15)## 2431:

the 33-gonal Number 33gon(13)## 2437:

Centered Hexagonal Number CH(29)## 2445:

the Truncated Tetrahedral Number TruTet(9)## 2449:

the 2449th Triangular Number is also a Dodecagonal Number## 2454:

the 39-gonal Number 39gon(12)## 2461:

Centered Triangle CT(41)## 2465:

Rhombic Dodecahedral Number RhoDod(9), also Octagonal Number O(29), also the 20-gonal Number 20gon(17)## 2466:

the 18-gonal Number 18gon(18), also the 18-gonal Jonathan Number 18gon(18gon(2))## 2470:

the Square Pyramidal Number SPyr(19)## 2471:

the 29-gonal Number 29gon(14)## 2481:

the Centered Pentagonal Number CP(32)## 2484:

the Nonagonal Number N(27), also the Heterogeneous Jonathan Number N(C(3))## 2485:

the Triangular Number T(70)## 2490:

the 15-gonal Number 15gon(20)## 2500:

a Square, S(50)## 2501:

a Pentagonal Number P(41), also the 5th Semiprime Pentagonal Number (41 x 61), also squares for first pair of digits, third, and fourth digit## 2508:

Hendecagonal Number, or 11-gonal number, Hen(24)## 2509:

the 34-gonal Number 34gon(13)## 2512:

the Heptagonal Number Hep(32), note that the index (32), the first digit (2) and the last 3 digits (512) are all powers of 2.## 2520:

the 40-gonal Number 40gon(12)## 2521:

a Centered Square Number CS(36)## 2535:

the 26-gonal Number, 26gon(15)## 2536:

the 23-gonal Number 23gon(16), also the Heterogeneous Jonathan Number 23gon(S(4))## 2541:

the 14-gonal Number 14gon(21)## 2556:

the Triangular Number T(71)## 2560:

a Constructable Number because (2^9)x5 (power of two times product of Fermat Primes)## 2562:

the 30-gonal Number 30gon(14)## 2563:

the 13-gonal Number 13gon(22)## 2570:

a Constructable Number because 2x5x257, a power of two times a product of Fermat Primes## 2584:

the square root of a Square Dodecagonal Number, also Centered Triangle CT(42), also the 17-gonal Number 17gon(19)## 2586:

the 41-gonal Number 41gon(12)## 2587:

the 35-gonal Number 35gon(13)## 2600:

the Tetrahedral Number Tet(24)## 2601:

the Pentagonal Pyramidal Number PPyr(17), also a Square, S(51), also the 21-gonal Number 21gon(17)## 2611:

Centered Hexagonal Number CH(30)## 2619:

the 19-gonal Number 19gon(18)## 2625:

a Pentagonal Number P(42)## 2626:

the Decagonal Number D(26), the Decagonal Number repeats the index twice (base 10)## 2628:

the Triangular Number T(72)## 2640:

Octagonal Number O(30), also the 27-gonal Number, 27gon(15)## 2641:

the Centered Pentagonal Number CP(31)## 2651:

the Stella Octangula Number StOct(11)## 2652:

the 42-gonal Number 42gon(12)## 2653:

the 31-gonal Number 31gon(14)## 2656:

the 24-gonal Number 24gon(16), also the Heterogeneous Jonathan Number 24gon(S(4))## 2665:

a Centered Square Number CS(37), also the 36-gonal Number 36gon(13)## 2673:

the Heptagonal Number Hep(33)## 2674:

the Nonagonal Number N(28)## 2680:

the 16-gonal Number 16gon(20)## 2701:

the Triangular Number T(73), also the 6th Semiprime Hexagonal Number (37 x 73)## 2704:

a Square, S(52)## 2710:

Centered Triangle CT(43)## 2718:

the 43-gonal Number, 43gon(12)## 2720:

a Constructable Number because (2^5)x5x17, a power of two times a product of Fermat Primes## 2725:

Hendecagonal Number, or 11-gonal number, Hen(25), hence the Heterogeneous Jonathan Number Hen(S(5))## 2736:

Octahedral Number Octh(16)## 2737:

the 22-gonal Number 22gon(17)## 2743:

the 37-gonal Number 37gon(13)## 2744:

the Cube C(14), also the 32-gonal Number 32gon(14)## 2745:

the 28-gonal Number 28gon(15)## 2751:

the 15-gonal Number 15gon(21)## 2752:

a Pentagonal Number P(43)## 2755:

the 18-gonal Number 18gon(19)## 2772:

the 20-gonal Number 20gon(18), also a Palindrome (base 10)## 2775:

the Triangular Number T(74)## 2776:

the 25-gonal number, 25gon(16), also the Heterogeneous Jonathan Number, 25gon(S(4))## 2784:

the 44-gonal Number 44gon(12)## 2791:

Centered Hexagonal Number CH(31)## 2806:

the Centered Pentagonal Number CP(32), also the 13-gonal Number 13gon(23)## 2809:

a Square, S(53)## 2813:

a Centered Square Number CS(38)## 2821:

Octagonal Number O(31), also the 38-gonal Number 38gon(13)## 2835:

the Decagonal Number D(27), also the Heterogeneous Jonathan Number D(C(3)); also the 33-gonal Number 33gon(14)## 2839:

Centered Triangle CT(44), also the Heptagonal Number Hep(34), also the Heptagonal Jonathan Number Hep(Hep(4)), also the 10th Semiprime Heptagonal Number (17 x 167)## 2850:

the Triangular Number T(75), also the 29-gonal Number 29gon(15)## 2856:

the Hexagonal Pyramidal Number HPyr(16)## 2870:

the Square Pyramidal Number SPyr(20), also the 17-gonal Number 17gon(20)## 2871:

the Nonagonal Number N(29)## 2873:

the 23-gonal Number 23gon(17)## 2882:

a Palindromic Pentagonal Number with Palindromic Index: P(44)## 2894:

the 14-gonal Number 14gon(22)## 2896:

the 26-gonal Number, 26gon(16), also the Heterogeneous Jonathan Number 26gon(S(4))## 2899:

the 39-gonal Number 39gon(13)## 2916:

a Square, S(54)## 2920:

the Heptagonal Pyramidal Number HepPyr(15)## 2925:

the Tetrahedral Number Tet(25), also the 21-gonal Number 21gon(18)## 2926:

the Triangular Number T(76), also the 19-gonal Number 19gon(19), also the 19-gonal Jonathan Number 19gon(19gon(2)); also the 34-gonal Number 34gon(14)## 2951:

Hendecagonal Number, or 11-gonal number, Hen(26)## 2955:

the 30-gonal Number 30gon(15)## 2961:

the 16-gonal Number 16gon(21)## 2965:

a Centered Square Number CS(39)## 2971:

Centered Triangle CT(45) {to be done: insert additional numbers in this sequence}## 2976:

the Centered Pentagonal Number CP(33)## 2977:

Centered Hexagonal Number CH(32); also the 40-gonal Number 40gon(13)## 3003:

the Triangular Number T(77)## 3008:

Octagonal Number O(32)## 3009:

the 24-gonal Number 24gon(17)## 3010:

the Heptagonal Number Hep(35)## 3015:

a Pentagonal Number P(45), also with first two digits double its last two## 3016:

the 27-gonal Number, 27gon(16), also the Heterogeneous Jonathan Number 27gon(S(4))## 3017:

the 35-gonal Number 35gon(14)## 3025:

a Square, S(55), also the Heterogeneous Jonathan Number S(T(10)), also the Heterogeneous Jonathan Number S(T(T(4))); also the 15-gonal Number 15gon(22)## 3052:

the Decagonal Number D(28)## 3055:

the 41-gonal Number 41gon(13)## 3059:

a Centered Cube Number CC(12), also the 14-gonal Number 14gon(23)## 3060:

the 13-gonal Number 13gon(24), also the 18-gonal Number 18gon(20), also the Pentatope Number Ptop(15), also the Pentatope Jonathan Number Ptop(Ptop(3)); also the 31-gonal Number 31gon(15)## 3072:

a Constructable Number because (2^10)x3 (power of two times product of Fermat Primes)## 3075:

the Nonagonal Number N(30)## 3078:

the Pentagonal Pyramidal Number PPyr(18), also the Pentagonal Pyramidal Jonathan Number PPyr(PPyr(3)); also the 22-gonal Number 22gon(18)## 3081:

the Triangular Number T(78), hence the Triangular Jonathan Number T(T(12))## 3084:

a Constructable Number because (2^2)x3x257, a power of two times a product of Fermat Primes## 3097:

the 20-gonal Number 20gon(19)## 3108:

the 36-gonal Number 36gon(14)## 3121:

a Centered Square Number CS(40)## 3133:

the 42-gonal Number 42gon(13), middle 2 digits equal index (base 10)## 3136:

a Square, S(56), also the 28-gonal Number 28gon(16), also the Heterogeneous Jonathan Number 28gon(S(4))## 3145:

the 25-gonal number, 25gon(17)## 3151:

a Pentagonal Number P(46), also the 6th Semiprime Pentagonal Number (23 x 137)## 3160:

the Triangular Number T(79)## 3165:

the 32-gonal Number 32gon(15)## 3169:

Centered Hexagonal Number CH(33)## 3171:

the 17-gonal Number 17gon(21)## 3186:

Hendecagonal Number, or 11-gonal number, Hen(27), also the Heterogeneous Jonathan Number Hen(C(3)); also the Heptagonal Number Hep(36), note that the two digits of the index (36) are also the first and last digit of the Heptagonal Number (base 10)## 3199:

the 37-gonal Number 37gon(14)## 3201:

Octagonal Number O(33)## 3211:

the 43-gonal Number, 43gon(13)## 3231:

the 23-gonal Number 23gon(18)## 3240:

the Triangular Number T(80)## 3249:

a Square, S(57)## 3250:

the 19-gonal Number 19gon(20)## 3256:

the 16-gonal Number 16gon(22), also the 29-gonal Number 29gon(16), also the Heterogeneous Jonathan Number 29gon(S(4))## 3264:

a Constructable Number because (2^6)x3x17, a power of two times a product of Fermat Primes## 3268:

the 21-gonal Number 21gon(19)## 3277:

the Decagonal Number D(29), also the 6th Semiprime Decagonal Number (29 x 113)## 3281:

the 26-gonal Number, 26gon(17)## 3286:

the Nonagonal Number N(31)## 3289:

the 44-gonal Number 44gon(13)## 3290:

a Pentagonal Number P(47), also the 38-gonal Number 38gon(14)## 3270:

the 33-gonal Number 33gon(15)## 3276:

the Tetrahedral Number Tet(26)## 3281:

Octahedral Number Octh(17) also a Centered Square Number CS(41)## 3290:

a Pentagonal Number## 3311:

the Square Pyramidal Number SPyr(21)## 3312:

the 15-gonal Number 15gon(23)## 3321:

the Triangular Number T(81), hence the Heterogeneous Jonathan Number T(S(9)), hence the Heterogeneous Jonathan Number T(S(S(3)))## 3325:

the 13-gonal Number 13gon(25), also the Heterogeneous Jonathan Number 13gon(S(5))## 3336:

the 14-gonal Number 14gon(24)## 3364:

a Square, S(58)## 3367:

Centered Hexagonal Number CH(34), also the Heptagonal Number Hep(37)## 3375:

the Cube C(15), also the 34-gonal Number 34gon(15)## 3376:

the 30-gonal Number 30gon(16), also the Heterogeneous Jonathan Number 30gon(S(4))## 3381:

the 18-gonal Number 18gon(21); also the 39-gonal Number 39gon(14)## 3384:

the 24-gonal Number 24gon(18)## 3400:

the Truncated Tetrahedral Number TruTet(10), also Octagonal Number O(34)## 3403:

the Triangular Number T(82)## 3417:

the Hexagonal Pyramidal Number HPyr(17) (note that the index 17 divides the first two digits and last two digits), also the 27-gonal Number, 27gon(17)## 3430:

Hendecagonal Number, or 11-gonal number, Hen(28)## 3432:

a Pentagonal Number P(48), and the 7th Central Binomial Coefficient## 3439:

Rhombic Dodecahedral Number RhoDod(10), also the 22-gonal Number 22gon(19)## 3440:

the 20-gonal Number 20gon(20), also the 20-gonal Jonathan Number 20gon(20gon(2))## 3444:

the Stella Octangula Number StOct(12)## 3445:

a Centered Square Number CS(42)## 3472:

the 40-gonal Number 40gon(14)## 3480:

the 35-gonal Number 35gon(15)## 3481:

a Square, S(59)## 3486:

the Triangular Number T(83)## 3487:

the 17-gonal Number 17gon(22)## 3496:

the 31-gonal Number 31gon(16), also the Heterogeneous Jonathan Number 31gon(S(4))## 3504:

the Nonagonal Number N(32)## 3510:

the Decagonal Number D(30)## 3536:

the Heptagonal Pyramidal Number HepPyr(16)## 3537:

the 25-gonal number, 25gon(18)## 3553:

the Heptagonal Number Hep(38), also the 28-gonal Number 28gon(17), also a Palindrome## 3563:

the 41-gonal Number 41gon(14)## 3565:

the 16-gonal Number 16gon(23)## 3570:

the Triangular Number T(84)## 3571:

Centered Hexagonal Number CH(35)## 3577:

a Pentagonal Number P(49), also the Heterogeneous Jonathan Number P(S(7)), and a Kaprekar Constant in base 2## 3585:

the 36-gonal Number 36gon(15)## 3591:

the 19-gonal Number 19gon(21)## 3600:

a Square, S(60)## 3601:

the 13-gonal Number 13gon(26)## 3605:

Octagonal Number O(35)## 3610:

the Pentagonal Pyramidal Number PPyr(19), also the 23-gonal Number 23gon(19)## 3612:

the 15-gonal Number 15gon(24)## 3613:

a Centered Square Number CS(43)## 3616:

the 32-gonal Number 32gon(16), also the Heterogeneous Jonathan Number 32gon(S(4))## 3625:

the 14-gonal Number 14gon(25), also the Heterogeneous Jonathan Number 14gon(S(5))## 3630:

the 21-gonal Number 21gon(20)## 3654:

the Tetrahedral Number Tet(27), also the Heterogeneous Jonathan Number Tet(C(3)); also the 42-gonal Number 42gon(14)## 3655:

the Triangular Number T(85)## 3683:

Hendecagonal Number, or 11-gonal number, Hen(29)## 3689:

the 29-gonal Number 29gon(17)## 3690:

the 26-gonal Number 26gon(18), also the 37-gonal Number 37gon(15)## 3718:

the 18-gonal Number 18gon(22)## 3721:

a Square, S(61)## 3725:

a Pentagonal Number P(50)## 3729:

the Nonagonal Number N(33)## 3736:

the 33-gonal Number 33gon(16), also Heterogeneous Jonathan Number 33gon(S(4))## 3741:

the Triangular Number T(86)## 3744:

the Heptagonal Number Hep(39)## 3745:

the 43-gonal Number, 43gon(14)## 3751:

the Decagonal Number D(31)## 3781:

Centered Hexagonal Number CH(36), also the 24-gonal Number 24gon(19)## 3785:

a Centered Square Number CS(44)## 3795:

the Square Pyramidal Number SPyr(22), also the 38-gonal Number 38gon(15)## 3801:

the 20-gonal Number 20gon(21)## 3816:

Octagonal Number O(36), also Heterogeneous Jonathan Number O(S(6))## 3818:

the 17-gonal Number 17gon(23)## 3820:

the 22-gonal Number 22gon(20)## 3825:

the 30-gonal Number 30gon(17)## 3828:

the Triangular Number T(87)## 3836:

the 44-gonal Number 44gon(14)## 3840:

a Constructable Number because (2^8)x3x5 (power of two times product of Fermat Primes)## 3843:

the 27-gonal Number, 27gon(18)## 3844:

a Square, S(62)## 3855:

a Constructable Number because 3x5x257, a power of two (here 2^0=1) times a product of Fermat Primes## 3856:

the 34-gonal Number 34gon(16), also the Heterogeneous Jonathan Number 34gon(S(4))## 3876:

the Pentatope Number Ptop(16) also a Pentagonal Number P(51)## 3888:

the 13-gonal Number 13gon(27), also the Heterogeneous Jonathan Number 13gon(C(3)); also the 16-gonal Number 16gon(24)## 3894:

Octahedral Number Octh(18)## 3900:

the 39-gonal Number 39gon(15)## 3916:

the Triangular Number T(88)## 3925:

a Centered Cube Number CC(13), also the 15-gonal Number 15gon(25), also the Heterogeneous Jonathan Number 15gon(S(5))## 3926:

the 14-gonal Number 14gon(26)## 3940:

the Heptagonal Number Hep(40)## 3945:

Hendecagonal Number, or 11-gonal number, Hen(30)## 3949:

the 19-gonal Number 19gon(22)## 3952:

the 25-gonal number, 25gon(19)## 3961:

a Centered Square Number CS(45), also the Nonagonal Number N(34), also the the 5th Semiprime Nonagonal Number (17 x 223), also the 31-gonal Number 31gon(17)## 3969:

a Square, S(63)## 3976:

the 35-gonal Number 35gon(16), also the Heterogeneous Jonathan Number 35gon(S(4))## 3996:

the 28-gonal Number 28gon(18)## 3997:

Centered Hexagonal Number CH(37), also a Centered Hexagonal Jonathan Number CH(CH(4))## 4000:

the Decagonal Number D(32)## 4005:

the Triangular Number T(89); also the 40-gonal Number 40gon(15)## 4010:

the 23-gonal Number 23gon(20)## 4011:

the 21-gonal Number 21gon(21), also the 21-gonal Jonathan Number 21gon(21gon(2))## 4030:

a Pentagonal Number P(52), also an an Abundant Number that is not the sum of some subset of its divisors## 4033:

the Truncated Octahedral Number TruOct(7), also Octagonal Number O(37), also the 7th Semiprime Octagonal Number (37 x 109)## 4047:

the Hexagonal Pyramidal Number HPyr(18)## 4060:

the Tetrahedral Number Tet(28)## 4071:

the 18-gonal Number 18gon(23)## 4080:

a Constructable Number because (2^4)x3x5x17, a power of two times a product of Fermat Primes## 4095:

the Triangular Number T(90)## 4096:

the Cube C(16), since 16 is itself a 4th power, this makes 4096 a 12th power, 2^12 and is also the Square S(64); also the 36-gonal Number 36gon(16), also the Heterogeneous Jonathan Number 36gon(S(4))## 4097:

the 32-gonal Number 32gon(17)## 4110:

the 41-gonal Number 41gon(15)## 4112:

a Constructable Number because (2^4)x257, a power of two times a product of Fermat Primes## 4123:

the 26-gonal Number, 26gon(19)## 4141:

a Centered Square Number CS(46), also the Heptagonal Number Hep(41), note that (base 10) the index (41) appears as the first two and last two digits of the Heptagonal Number, also the 11th Semiprime Heptagonal Number (41 x 101)## 4149:

the 29-gonal Number 29gon(18)## 4164:

the 17-gonal Number 17gon(24)## 4180:

the 20-gonal Number 20gon(22)## 4186:

the Triangular Number T(91), hence the Triangular Jonathan Number T(T(13)); also the 13-gonal Number 13gon(28)## 4187:

a Pentagonal Number P(53), and the smallest Rabin-Miller Pseudoprime with an odd reciprocal period## 4200:

the Pentagonal Pyramidal Number PPyr(20), also the Nonagonal Number N(35), also the 24-gonal Number 24gon(20)## 4215:

the 42-gonal Number 42gon(15), number ends with 2-digit index (base 10)## 4216:

Hendecagonal Number, or 11-gonal Number Hen(31), also the 37-gonal Number 37gon(16), also the Heterogeneous Jonathan Number 37gon(S(4))## 4219:

Centered Hexagonal Number CH(38)## 4221:

the 22-gonal Number 22gon(21), last 2 digits same as index (base 10)## 4225:

a Square, S(65); also the 16-gonal Number 16gon(25), also the Heterogeneous Jonathan Number 16gon(S(5))## 4233:

the Heptagonal Pyramidal Number HepPyr(17), also the 33-gonal Number 33gon(17)## 4239:

the 14-gonal Number 14gon(27), also the Heterogeneous Jonathan Number 14gon(C(3))## 4251:

the 15-gonal Number 15gon(26)## 4256:

Octagonal Number O(38)## 4257:

the Decagonal Number D(33)## 4278:

the Triangular Number T(92)## 4294:

the 27-gonal Number, 27gon(19)## 4302:

the 30-gonal Number 30gon(18)## 4320:

the 43-gonal Number, 43gon(15)## 4324:

the Square Pyramidal Number SPyr(23), also the 19-gonal Number 19gon(23)## 4325:

a Centered Square Number CS(47) {to be done: append those Centered Squares following in the sequence}## 4336:

the 38-gonal Number 38gon(16), also the Heterogeneous Jonathan Number 38gon(S(4))## 4347:

a Pentagonal Heptagonal Number: the Pentagonal Number P(54), also the Heptagonal Number Hep(42)## 4352:

a Constructable Number because (2^8)x17 (power of two times product of Fermat Primes)## 4356:

a Square, S(66), also the Heterogeneous Jonathan Number S(T(11))## 4369:

the 34-gonal Number 34gon(17), also a Constructable Number because 17x257, a power of two (here 2^0=1) times a product of Fermat Primes## 4371:

the Triangular Number T(93)## 4381:

the Stella Octangula Number StOct(13)## 4390:

the 25-gonal number, 25gon(20)## 4411:

the 21-gonal Number 21gon(22)## 4421:

1! - 2! + 3! - 4! + 5! - 6! + 7!## 4425:

the 44-gonal Number 44gon(15)## 4431:

the 23-gonal Number 23gon(21)## 4440:

the 18-gonal Number 18gon(24)## 4446:

the Nonagonal Number N(36), also the Heterogeneous Jonathan Number N(S(6))## 4447:

Centered Hexagonal Number CH(39)## 4455:

the 31-gonal Number 31gon(18)## 4456:

the 39-gonal Number 39gon(16), also the Heterogeneous Jonathan Number 39gon(S(4))## 4465:

the Triangular Number T(94), also the 28-gonal Number 28gon(19)## 4485:

Octagonal Number O(39)## 4489:

a Square, S(67)## 4495:

the Tetrahedral Number Tet(29), also the 13-gonal Number 13gon(29)## 4496:

Hendecagonal Number, or 11-gonal number, Hen(32)## 4505:

the 35-gonal Number 35gon(17)## 4510:

a Pentagonal Number P(55)## 4522:

the Decagonal Number D(34)## 4525:

the 17-gonal Number 17gon(25), also the Heterogeneous Jonathan Number 17gon(S(5))## 4558:

the Heptagonal Number Hep(43)## 4560:

the Triangular Number T(95)## 4564:

the 14-gonal Number 14gon(28)## 4576:

the Truncated Tetrahedral Number TruTet(11), also the 16-gonal Number 16gon(26); also the 40-gonal Number 40gon(16), also the Heterogeneous Jonathan Number 40gon(S(4))## 4577:

the 20-gonal Number 20gon(23)## 4579:

Octahedral Number Octh(19), Octahedral Jonathan Number Octh(Octh(3))## 4580:

the 26-gonal Number, 26gon(20)## 4590:

the 15-gonal Number 15gon(27), also the Heterogeneous Jonathan Number 15gon(C(3))## 4608:

the 32-gonal Number 32gon(18)## 4624:

a Square, S(68)## 4636:

the 29-gonal Number 29gon(19)## 4641:

Rhombic Dodecahedral Number RhoDod(11), also the 24-gonal Number 24gon(21), also the 36-gonal Number 36gon(17)## 4642:

the 22-gonal Number 22gon(22), also the 22-gonal Jonathan Number 22gon(22gon(3))## 4656:

the Triangular Number T(96)## 4676:

a Pentagonal Number P(56), and the sum of the first seven 4th powers## 4681:

Centered Hexagonal Number CH(40)## 4696:

the 41-gonal Number 41gon(16), also the Heterogeneous Jonathan Number 41gon(S(4))## 4699:

the Nonagonal Number N(37), also the the 6th Semiprime Nonagonal Number (37 x 127)## 4716:

the 19-gonal Number 19gon(24)## 4720:

Octagonal Number O(40), also Octagonal Jonathan Number O(O(4))## 4750:

the Hexagonal Pyramidal Number HPyr(19)## 4753:

the Triangular Number T(97)## 4761:

a Square, S(69); also the 33-gonal Number 33gon(18)## 4770:

the 27-gonal Number, 27gon(20)## 4774:

the Heptagonal Number Hep(44), both index and Heptagonal Number are Palindromes (in base 10)## 4777:

the 37-gonal Number 37gon(17)## 4785:

Hendecagonal Number, or 11-gonal number, Hen(33)## 4795:

the Decagonal Number D(35)## 4807:

the 30-gonal Number 30gon(19)## 4815:

the 13-gonal Number 13gon(30)## 4816:

the 42-gonal Number 42gon(16), number ends with 2-digit index (base 10); also the Heterogeneous Jonathan Number 42gon(S(4))## 4825:

the 18-gonal Number 18gon(25), hence the Heterogeneous Jonathan Number 18gon(S(5))## 4830:

the 21-gonal Number 21gon(23)## 4845:

the Pentatope Number Ptop(17) also a Pentagonal Number P(57), and 20C4## 4851:

the Pentagonal Pyramidal Number PPyr(21), also the Triangular Number T(98), also the 25-gonal number, 25gon(21)## 4873:

the 23-gonal Number 23gon(22)## 4900:

the Square Pyramidal Number SPyr(24), and the Square S(70), this was proven by Jud McCranie to be the only Square Pyramidal Number which is also a Square## 4901:

the 14-gonal Number 14gon(29), also the 17-gonal Number 17gon(26)## 4913:

the Cube C(17), also the 38-gonal Number 38gon(17)## 4914:

the 34-gonal Number 34gon(18)## 4921:

Centered Hexagonal Number CH(41)## 4936:

the 43-gonal Number, 43gon(16), 43-gonal index a square, 1st two digits a square, last 2 digits a square (base 10); also the Heterogeneous Jonathan Number 43gon(S(4))## 4941:

a Centered Cube Number CC(14), also the 16-gonal Number 16gon(27), hence the Heterogeneous Jonathan Number 16gon(C(3))## 4942:

the 15-gonal Number 15gon(28)## 4950:

the Triangular Number T(99)## 4959:

the Nonagonal Number N(38)## 4960:

the Tetrahedral Number Tet(30), also the 28-gonal Number 28gon(20)## 4961:

Octagonal Number O(41)## 4978:

the 31-gonal Number 31gon(19)## 4992:

the 20-gonal Number 20gon(24)## 4995:

the Heptagonal Number Hep(45), note that the two digits of index (45) are the first and last digit of the Heptagonal Number (base 10)## 5000:

Five Thousand One Hundred Four is the largest integer whose English name repeats no letter. See Numbers in Recreational Linguistics## 5016:

the Heptagonal Pyramidal Number HepPyr(18)## 5017:

a Pentagonal Number P(58), and both 5 and 17 are Fermat Primes## 5041:

a Square, S(71)## 5049:

the 39-gonal Number 39gon(17)## 5050:

the Triangular Number T(100), hence the Heterogeneous Jonathan Number T(S(10)) (first two digits same as last two digits)## 5056:

the 44-gonal Number 44gon(16), also the Heterogeneous Jonathan Number 44gon(S(4))## 5061:

the 26-gonal Number, 26gon(21)## 5067:

the 35-gonal Number 35gon(18)## 5076:

the Decagonal Number D(36), also the Heterogeneous Jonathan Number D(S(6))## 5083:

Hendecagonal Number, or 11-gonal number, Hen(34); also the 22-gonal Number 22gon(23)## 5104:

the 24-gonal Number 24gon(22)## 5120:

a Constructable Number because (2^10)x5 (power of two times product of Fermat Primes)## 5125:

the 19-gonal Number 19gon(25), hence the Heterogeneous Jonathan Number 19gon(S(5))## 5146:

the 13-gonal Number 13gon(31)## 5151:

the Triangular Number T(101) (first two digits same as last two digits)## 5140:

a Constructable Number because (2^2)x5x257, a power of two times a product of Fermat Primes## 5149:

the 32-gonal Number 32gon(19)## 5150:

the 29-gonal Number 29gon(20)## 5167:

Centered Hexagonal Number CH(42) also, 1! + 3! + 5! + 7!## 5184:

a Square, S(72)## 5185:

the 40-gonal Number 40gon(17)## 5192:

a Pentagonal Number P(59)## 5208:

Octagonal Number O(42)## 5220:

the 36-gonal Number 36gon(18)## 5221:

the Heptagonal Number Hep(46), also the 12th Semiprime Heptagonal Number (23 x 227)## 5226:

the Nonagonal Number N(39), also the 18-gonal Number 18gon(26) which as a the 18-gonal Number ends with its index (base 10)## 5253:

the Triangular Number T(102)## 5268:

the 21-gonal Number 21gon(24)## 5271:

the 27-gonal Number, 27gon(21)## 5320:

the 16-gonal Number 16gon(28)## 5321:

the 41-gonal Number 41gon(17)## 5329:

a Square, S(73)## 5335:

the 25-gonal number, 25gon(22)## 5336:

the 23-gonal Number 23gon(23), also the 23-gonal Jonathan Number 23gon(23gon(2))## 5340:

Octahedral Number Octh(20), also the 30-gonal Number 30gon(20)## 5250:

the 14-gonal Number 14gon(30)## 5292:

the 17-gonal Number 17gon(27), also the Heterogeneous Jonathan Number 17gon(C(3))## 5307:

the 15-gonal Number 15gon(29)## 5320:

the 33-gonal Number 33gon(19)## 5356:

the Triangular Number T(103)## 5365:

the Decagonal Number D(37)## 5370:

the Pentagonal Number P(__)## 5373:

the 37-gonal Number 37gon(18)## 5390:

the Hendecagonal Number, or 11-gonal Number Hen(34)## 5419:

Centered Hexagonal Number CH(43)## 5530:

the Hexagonal Pyramidal Number HPyr(20), also the 31-gonal Number 31gon(20)## 5370:

a Pentagonal Number P(60)## 5390:

Hendecagonal Number, or 11-gonal number, Hen(35)## 5425:

the 20-gonal Number 20gon(25), also the Heterogeneous Jonathan Number 20gon(S(5))## 5440:

a Constructable Number because (2^6)x5x17, a power of two times a product of Fermat Primes (which is not the basis of the slogan in American History: 5440 or Fight!")## 5452:

the Heptagonal Number Hep(47)## 5456:

the Tetrahedral Number Tet(31)## 5457:

the 42-gonal Number 42gon(17)## 5460:

the Triangular Number T(104)## 5461:

Octagonal Number O(43), also the 8th Semiprime Octagonal Number (43 x 127)## 5474:

the Stella Octangula Number StOct(14), also the Stella Octangula Jonathan Number StOct(StOct(2))## 5476:

a Square, S(74)## 5481:

the 28-gonal Number 28gon(21)## 5488:

the 13-gonal Number 13gon(32)## 5491:

the 34-gonal Number 34gon(19)## 5500:

the Nonagonal Number N(40)## 5525:

the Square Pyramidal Number SPyr(25)## 5526:

the 38-gonal Number 38gon(18)## 5544:

the 22-gonal Number 22gon(24)## 5551:

a Pentagonal Number P(61), also## 5125:

the 19-gonal Number 19gon(26)## 5565:

the Triangular Number T(105), hence the Triangular Jonathan Number T(T(14))## 5566:

the Pentagonal Pyramidal Number PPyr(22), also the 26-gonal Number, 26gon(22)## 5589:

the 24-gonal Number 24gon(23)## 5593:

the 43-gonal Number, 43gon(17)## 5611:

the 14-gonal Number 14gon(31)## 5625:

a Square, S(75)## 5643:

the 18-gonal Number 18gon(27), also the Heterogeneous Jonathan Number 18gon(C(3))## 5662:

the Decagonal Number D(38), also the 35-gonal Number 35gon(19)## 5671:

the Triangular Number T(106)## 5677:

Centered Hexagonal Number CH(44)## 5679:

the 39-gonal Number 39gon(18)## 5685:

the 15-gonal Number 15gon(30)## 5688:

the Heptagonal Number Hep(48)## 5691:

the 29-gonal Number 29gon(21)## 5706:

Hendecagonal Number, or 11-gonal number, Hen(36), also the Heterogeneous Jonathan Number Hen(S(6))## 5713:

the 16-gonal Number 16gon(29)## 5720:

Octagonal Number O(44), also the 32-gonal Number 32gon(20)## 5725:

the 21-gonal Number 21gon(25), also the Heterogeneous Jonathan Number 21gon(S(5))## 5729:

the 44-gonal Number 44gon(17)## 5735:

a Pentagonal Number P(62)## 5778:

the Triangular Number T(107), also a Square, S(76)## 5781:

the Nonagonal Number N(41)## 5797:

the 27-gonal Number, 27gon(22)## 5820:

the 23-gonal Number 23gon(24)## 5832:

the Cube C(18); also the 40-gonal Number 40gon(18)## 5833:

the 36-gonal Number 36gon(19)## 5841:

the 13-gonal Number 13gon(33)## 5842:

the 25-gonal number, 25gon(23)## 5876:

the 20-gonal Number 20gon(26)## 5886:

the Triangular Number T(108)## 5890:

the Heptagonal Pyramidal Number HepPyr(19)## 5901:

the 30-gonal Number 30gon(21)## 5910:

the 33-gonal Number 33gon(20)## 5922:

a Pentagonal Number P(63)## 5929:

a Square, S(77), also the Heptagonal Number Hep(49), also the Heterogeneous Jonathan Number Hep(S(7))## 5941:

Centered Hexagonal Number CH(45)## 5967:

the Decagonal Number D(39)## 5984:

the Tetrahedral Number Tet(32), also the 14-gonal Number 14gon(32)## 5985:

the Pentatope Number Ptop(18), also Octagonal Number O(45), also the 41-gonal Number 41gon(18)## 5994:

the 19-gonal Number 19gon(27), hence the Heterogeneous Jonathan Number 19gon(C(3))## 5995:

the Triangular Number T(109), and a Palindrome## 5996:

the Truncated Tetrahedral Number TruTet(12)## 5698:

the 17-gonal Number 17gon(28)## 6004:

the 37-gonal Number 37gon(19)## 6025:

the 22-gonal Number 22gon(25), hence the Heterogeneous Jonathan Number 22gon(S(5))## 6028:

the 28-gonal Number 28gon(22)## 6031:

Hendecagonal Number, or 11-gonal number, Hen(37)## 6069:

the Nonagonal Number N(42)## 6076:

the 15-gonal Number 15gon(31), also the 18-gonal Number 18gon(28)## 6084:

a Square, S(78), also the Heterogeneous Jonathan Number S(T(12))## 6095:

Rhombic Dodecahedral Number RhoDod(12), also the 26-gonal Number, 26gon(23)## 6096:

the 24-gonal Number 24gon(24), also the 24-gonal Jonathan Number 24gon(24gon(2))## 6100:

the 34-gonal Number 34gon(20)## 6105:

the Triangular Number T(110)## 6111:

the 31-gonal Number 31gon(21)## 6112:

a Pentagonal Number P(64), also the Heterogeneous Jonathan Number P(S(8))## 6119:

a Centered Cube Number CC(15), also the 17-gonal Number 17gon(29), also the next year from now which is the same upside-down## 6120:

the 16-gonal Number 16gon(30)## 6138:

the 42-gonal Number 42gon(18)## 6144:

a Constructable Number because (2^11)x3 (power of two times product of Fermat Primes)## 6168:

a Constructable Number because (2^3)x3x257, a power of two times a product of Fermat Primes## 6175:

the Heptagonal Number Hep(50), also the 38-gonal Number 38gon(19)## 6181:

Octahedral Number Octh(21)## 6201:

the Square Pyramidal Number SPyr(26), also the 21-gonal Number 21gon(26)## 6205:

the 13-gonal Number 13gon(34)## 6211:

Centered Hexagonal Number CH(46)## 6216:

the Triangular Number T(111)## 6241:

a Square, S(79)## 6256:

Octagonal Number O(46)## 6259:

the 29-gonal Number 29gon(22)## 6266:

the Truncated Octahedral Number TruOct(8)## 6280:

the Decagonal Number D(40)## 6290:

the 35-gonal Number 35gon(20)## 6291:

the 43-gonal Number, 43gon(18)## 6305:

a Pentagonal Number P(65)## 6321:

the 32-gonal Number 32gon(21), last 2 digits same as index (base 10), also in base 10: 6=3x2x1## 6325:

the 23-gonal Number 23gon(25), also the Heterogeneous Jonathan Number 23gon(S(5))## 6328:

the Triangular Number T(112)## 6345:

the 20-gonal Number 20gon(27), also the Heterogeneous Jonathan Number 20gon(C(3))## 6346:

the 39-gonal Number 39gon(19)## 6348:

the Pentagonal Pyramidal Number PPyr(23), also the 27-gonal Number, 27gon(23)## 6364:

the Nonagonal Number N(43)## 6365:

Hendecagonal Number, or 11-gonal number, Hen(38)## 6369:

the 14-gonal Number 14gon(33)## 6372:

the 25-gonal number, 25gon(24)## 6391:

the Hexagonal Pyramidal Number HPyr(21)## 6400:

a Square, S(80)## 6426:

the Heptagonal Number Hep(51)## 6441:

the Triangular Number T(113)## 6444:

the 44-gonal Number 44gon(18)## 6454:

the 19-gonal Number 19gon(28)## 6480:

the 15-gonal Number 15gon(32), also the 36-gonal Number 36gon(20)## 6487:

Centered Hexagonal Number CH(47) {to be done: append additional numbers in this sequence}## 6490:

the 30-gonal Number 30gon(22)## 6501:

a Pentagonal Number P(66), and has a square whose reverse is also a square## 6517:

the 40-gonal Number 40gon(19)## 6525:

the 18-gonal Number 18gon(29)## 6526:

the 22-gonal Number 22gon(26)## 6528:

a Constructable Number because (2^7)x3x17, a power of two times a product of Fermat Primes## 6531:

the 33-gonal Number 33gon(21)## 6533:

Octagonal Number O(47)## 6541:

the 16-gonal Number 16gon(31)## 6545:

the Tetrahedral Number Tet(33)## 6555:

the Triangular Number T(114), also the 17-gonal Number 17gon(30)## 6561:

a Square, S(81), hence a Biquadratic 9^4, hence an 8th power 3^8## 6580:

the 13-gonal Number 13gon(35)## 6601:

the Decagonal Number D(41), also the 28-gonal Number 28gon(23)## 6623:

the 6623rd Dodecagonal Number is also a Triangular Number## 6625:

the 24-gonal Number 24gon(25), also the Heterogeneous Jonathan Number 24gon(S(5))## 6648:

the 26-gonal Number, 26gon(24)## 6666:

the Nonagonal Number N(44), repdigit with repdigit index as Nonagonal## 6670:

the Triangular Number T(115), also the 37-gonal Number 37gon(20)## 6682:

the Heptagonal Number Hep(52)## 6688:

the 41-gonal Number 41gon(19)## 6696:

the 21-gonal Number 21gon(27), also the Heterogeneous Jonathan Number 21gon(C(3))## 6700:

a Pentagonal Number P(67)## 6708:

Hendecagonal Number, or 11-gonal number, Hen(39)## 6721:

the 31-gonal Number 31gon(22)## 6724:

a Square, S(82)## 6735:

the Stella Octangula Number StOct(15)## 6741:

the 34-gonal Number 34gon(21)## 6766:

the 14-gonal Number 14gon(34)## 6786:

the Triangular Number T(116)## 6816:

Octagonal Number O(48)## 6832:

the 20-gonal Number 20gon(28)## 6851:

the 23-gonal Number 23gon(26)## 6854:

the 29-gonal Number 29gon(23)## 6859:

the Cube C(19), also the 42-gonal Number 42gon(19)## 6860:

the Heptagonal Pyramidal Number HepPyr(20), also the 38-gonal Number 38gon(20)## 6889:

a Square, S(83)## 6897:

the 15-gonal Number 15gon(33)## 6902:

a Pentagonal Number P(68)## 6903:

the Triangular Number T(117)## 6924:

the 27-gonal Number, 27gon(24)## 6925:

the 25-gonal number, 25gon(25), also the Heterogeneous Jonathan Bumber 25gon(S(5))## 6930:

the Decagonal Number D(42), also the square root of a Square Triangular number (see Sloane A001110), and also the Square Pyramidal Number SPyr(27)## 6931:

the 19-gonal Number 19gon(29)## 6943:

the Heptagonal Number Hep(53)## 6951:

the 35-gonal Number 35gon(21)## 6952:

the 32-gonal Number 32gon(22)## 6966:

the 13-gonal Number 13gon(36), also the Heterogeneous Jonathan Number 13gon(S(6))## 6975:

the Nonagonal Number N(45)## 6976:

the 16-gonal Number 16gon(32)## 6990:

the 18-gonal Number 18gon(30)## 7006:

the 17-gonal Number 17gon(31)## 7056:

a Square, S(84)## 7021:

the Triangular Number T(118)## 7030:

the 43-gonal Number, 43gon(19)## 7047:

the 22-gonal Number 22gon(27), also the Heterogeneous Jonathan Number 22gon(C(3))## 7050:

the 39-gonal Number 39gon(20)## 7060:

Hendecagonal Number, or 11-gonal number, Hen(40)## 7105:

Octagonal Number O(49), also the Heterogeneous Jonathan Number O(S(7))## 7106:

Octahedral Number Octh(22)## 7107:

a Pentagonal Number P(69), also the 30-gonal Number 30gon(23)## 7140:

the Tetrahedral Number Tet(34), also the Triangular Number T(119)## 7146:

the 24-gonal Number 24gon(26)## 7161:

the 36-gonal Number 36gon(21)## 7175:

the 14-gonal Number 14gon(35)## 7183:

the 33-gonal Number 33gon(22)## 7200:

the Pentagonal Pyramidal Number PPyr(24), also the 28-gonal Number 28gon(24)## 7201:

the 44-gonal Number 44gon(19)## 7209:

the Heptagonal Number Hep(54)## 7210:

the 21-gonal Number 21gon(28)## 7225:

a Square, S(85), also the 26-gonal Number, 26gon(25), also the Heterogeneous Jonathan Number 26gon(S(5))## 7240:

the 40-gonal Number 40gon(20)## 7260:

the Triangular Number T(120), hence the Triangular Jonathan Number T(2,15) = T(T(15)), also the Triangular Jonathan Number T(3,5) = T(T(T(5))) also the Heterogeneous Jonathan Number T(Tet(8))## 7267:

the Decagonal Number D(43)## 7291:

the Nonagonal Number N(46), also Nonagonal Jonathan Number N(N(4))## 7315:

the Pentatope Number Ptop(19), also a Pentagonal Number P(70), and 22C4## 7327:

the 15-gonal Number 15gon(34)## 7337:

the Hexagonal Pyramidal Number HPyr(22), also the 20-gonal Number 20gon(29), also a Palindrome (base 10)## 7360:

the 31-gonal Number 31gon(23)## 7363:

the 13-gonal Number 13gon(37)## 7371:

the 37-gonal Number 37gon(21)## 7381:

the Triangular Number T(121), hence the Heterogeneous Jonathan Number T(S(11))## 7396:

a Square, S(86)## 7398:

the 23-gonal Number 23gon(27), also the Heterogeneous Jonathan Number 23gon(C(3))## 7400:

Octagonal Number O(50)## 7414:

the 34-gonal Number 34gon(22)## 7421:

Hendecagonal Number, or 11-gonal number, Hen(41)## 7425:

the 16-gonal Number 16gon(33), also the 19-gonal Number 19gon(30)## 7430:

the 41-gonal Number 41gon(20)## 7471:

a Centered Cube Number CC(16), also the 18-gonal Number 18gon(31)## 7472:

the 17-gonal Number 17gon(32)## 7476:

the 29-gonal Number 29gon(24)## 7480:

the Heptagonal Number Hep(55), also the Heptagonal Jonathan Number Hep(Hep(5))## 7501:

the 25-gonal number, 25gon(26)## 7503:

the Triangular Number T(122)## 7525:

the 27-gonal Number, 27gon(25), also the Heterogeneous Jonathan Number 27gon(S(5))## 7526:

a Pentagonal Number P(71)## 7569:

a Square, S(87)## 7581:

the 38-gonal Number 38gon(21)## 7588:

the 22-gonal Number 22gon(28)## 7596:

the 14-gonal Number 14gon(36), also the Heterogeneous Jonathan Number 14gon(S(6))## 7612:

the Decagonal Number D(44)## 7613:

the 32-gonal Number 32gon(23)## 7614:

the Nonagonal Number N(47)## 7620:

the 42-gonal Number 42gon(20)## 7626:

the Triangular Number T(123)## 7645:

the 35-gonal Number 35gon(22)## 7680:

a Constructable Number because (2^9)x3x5 (power of two times product of Fermat Primes)## 7683:

the Truncated Tetrahedral Number TruTet(13)## 7701:

Octagonal Number O(51)## 7710:

a Constructable Number because 2x3x5x257, a power of two times a product of Fermat Primes## 7714:

the Square Pyramidal Number SPyr(28)## 7740:

a Pentagonal Number P(72)## 7743:

the 21-gonal Number 21gon(29)## 7744:

a Square, S(88)## 7749:

the 24-gonal Number 24gon(27), also the Heterogeneous Jonathan Number 24gon(C(3))## 7750:

the Triangular Number T(124)## 7752:

the 30-gonal Number 30gon(24)## 7756:

the Heptagonal Number Hep(56)## 7770:

the Tetrahedral Number Tet(35), also the Tetrahedral Jonathan Number Tet(Tet(5)), also the 15-gonal Number 15gon(35)## 7771:

the 13-gonal Number 13gon(38)## 7791:

Hendecagonal Number, or 11-gonal number, Hen(42); also the 39-gonal Number 39gon(21)## 7810:

the 43-gonal Number, 43gon(20)## 7825:

Rhombic Dodecahedral Number RhoDod(13), also the 28-gonal Number 28gon(25), also Heterogeneous Jonathan Number 28gon(S(5))## 7826:

the 26-gonal Number, 26gon(26), base 10 the last two digits equal the index and the 26 of 26gon## 7860:

the 20-gonal Number 20gon(30)## 7866:

the 33-gonal Number 33gon(23)## 7875:

the Triangular Number T(125), hence the Heterogeneous Jonathan Number T(C(5))## 7876:

the 36-gonal Number 36gon(22)## 7888:

the 16-gonal Number 16gon(34)## 7921:

the 7921st Dodecagonal Number is also a Square, also 7921 itself is a Square, S(89)## 7931:

the Heptagonal Pyramidal Number HepPyr(21)## 7936:

the 19-gonal Number 19gon(31)## 7944:

the Nonagonal Number N(48)## 7953:

the 17-gonal Number 17gon(33)## 7957:

a Pentagonal Number P(73)## 7965:

the Decagonal Number D(45)## 7966:

the 23-gonal Number 23gon(28)## 7968:

the 18-gonal Number 18gon(32)## 8000:

the Cube C(20), also the 44-gonal Number 44gon(20)## 8001:

the Triangular Number T(126); also the 40-gonal Number 40gon(21)## 8008:

Octagonal Number O(52)## 8028:

the 31-gonal Number 31gon(24)## 8029:

the 14-gonal Number 14gon(37)## 8037:

the Heptagonal Number Hep(57)## 8100:

a Square, S(90), also the 25-gonal number, 25gon(27), also the Heterogeneous Jonathan Number 25gon(C(3))## 8107:

the 37-gonal Number 37gon(22)## 8119:

Octahedral Number Octh(23), also the 34-gonal Number 34gon(23)## 8125:

the Pentagonal Pyramidal Number PPyr(25), also the 29-gonal Number 29gon(25), also the Heterogeneous Jonathan Number 29gon(S(5))## 8128:

the Triangular Number T(127), also the Hexagonal Number H(64)## 8149:

the 22-gonal Number 22gon(29)## 8151:

the 27-gonal Number, 27gon(26)## 8160:

a Constructable Number because (2^5)x3x5x17, a power of two times a product of Fermat Primes## 8170:

Hendecagonal Number, or 11-gonal number, Hen(43) {to be done: insert additional numbers in this sequence}## 8176:

the Stella Octangula Number StOct(16)## 8177:

a Pentagonal Number P(74)## 8190:

the 13-gonal Number 13gon(39)## 8211:

the 41-gonal Number 41gon(21)## 8224:

a Constructable Number because (2^5)x257, a power of two times a product of Fermat Primes## 8226:

the 15-gonal Number 15gon(36), also the Heterogeneous Jonathan Number 15gon(S(6))## 8256:

the Triangular Number T(128)## 8281:

a Square, S(91), also the Heterogeneous Jonathan Number S(T(13)), also the Nonagonal Number N(49), also the Heterogeneous Jonathan Number N(S(7)), therefore a Heterogeneous Jonathan Number in 2 ways## 8295:

the 21-gonal Number 21gon(30)## 8304:

the 32-gonal Number 32gon(24)## 8321:

Octagonal Number O(53)## 8323:

the Heptagonal Number Hep(58)## 8326:

the Decagonal Number D(46)## 8338:

the 38-gonal Number 38gon(22), 38-gonal palindromic number starts with 38-reversed and ends with 38 (base 10)## 8344:

the 24-gonal Number 24gon(28)## 8365:

the 16-gonal Number 16gon(35)## 8372:

the Hexagonal Pyramidal Number HPyr(23), also the 35-gonal Number 35gon(23)## 8385:

the Triangular Number T(129), also the Hexagonal Number H(65)## 8400:

a Pentagonal Number P(75), and divisible by its reverse (base 10)## 8401:

the 20-gonal Number 20gon(31)## 8421:

the 42-gonal Number 42gon(21), number ends with 2-digit index (base 10)## 8425:

the 30-gonal Number 30gon(25), also the Heterogeneous Jonathan Number 30gon(S(5))## 8436:

the Tetrahedral Number Tet(36)## 8449:

the 17-gonal Number 17gon(34)## 8451:

the 26-gonal Number, 26gon(27), also the Heterogeneous Jonathan Number 26gon(C(3))## 8464:

a Square, S(92); also the 19-gonal Number 19gon(32)## 8474:

the 14-gonal Number 14gon(38)## 8476:

the 28-gonal Number 28gon(26)## 8481:

the 18-gonal Number 18gon(33)## 8515:

the Triangular Number T(130)## 8555:

the Square Pyramidal Number SPyr(29), also the 23-gonal Number 23gon(29)## 8558:

Hendecagonal Number, or 11-gonal number, Hen(44)## 8569:

the 39-gonal Number 39gon(22)## 8580:

the 33-gonal Number 33gon(24)## 8614:

the Heptagonal Number Hep(59)## 8620:

the 13-gonal Number 13gon(40)## 8625:

the Nonagonal Number N(50), also the 36-gonal Number 36gon(23)## 8626:

a Pentagonal Number P(76)## 8631:

the 43-gonal Number, 43gon(21)## 8640:

Octagonal Number O(54)## 8646:

the Triangular Number T(131), also also the Hexagonal Number H(66)## 8649:

a Square, S(93)## 8695:

the Decagonal Number D(47), also the 15-gonal Number 15gon(37)## 8704:

a Constructable Number because (2^9)x17 (power of two times product of Fermat Primes)## 8722:

the 25-gonal number, 25gon(28)## 8725:

the 31-gonal Number 31gon(25), also the Heterogeneous Jonathan Number 31gon(S(5))## 8730:

the 22-gonal Number 22gon(30)## 8738:

a Constructable Number because 2x17x257, a power of two times a product of Fermat Primes## 8778:

the Triangular Number T(132)## 8800:

the 40-gonal Number 40gon(22)## 8801:

the 29-gonal Number 29gon(26)## 8802:

the 27-gonal Number, 27gon(27), also the Heterogeneous Jonathan Number 27gon(C(3))## 8836:

a Square, S(94)## 8841:

the 44-gonal Number 44gon(21)## 8855:

the Pentatope Number Ptop(20) also a Pentagonal Number P(77), and 23C4## 8856:

the 16-gonal Number 16gon(36), also the Heterogeneous Jonathan Number 16gon(S(6)); also the 34-gonal Number 34gon(24)## 8866:

the 21-gonal Number 21gon(31)## 8878:

the 37-gonal Number 37gon(23)## 8910:

the Heptagonal Number Hep(60)## 8911:

a Carmichael Number; also a Triangular Number, and Dodecagonal Number the Triangular Number T(133), the Dodecagonal Number Dod(__), also the Hexagonal Number H(67)## 8920:

the Heptagonal Number Hep(60)## 8931:

the 14-gonal Number 14gon(39), also the 14gonal Jonathan Number 14gon(14gon(3))## 8955:

Hendecagonal Number, or 11-gonal number, Hen(45)## 8960:

the 17-gonal Number 17gon(35), also the 20-gonal Number 20gon(32)## 8961:

the 24-gonal Number 24gon(29)## 8965:

Octagonal Number O(55)## 8976:

the Nonagonal Number N(51)## 9009:

a Centered Cube Number CC(17), also the 19-gonal Number 19gon(33), also a Palindrome## 9010:

the 18-gonal Number 18gon(34)## 9025:

a Square, S(95), also the 32-gonal Number 32gon(25), also the Heterogeneous Jonathan Number 32gon(S(5))## 9031:

the 41-gonal Number 41gon(22)## 9045:

the Triangular Number T(134)## 9061:

the 13-gonal Number 13gon(41)## 9072:

the Decagonal Number D(48)## 9087:

a Pentagonal Number P(78)## 9100:

the 26-gonal Number, 26gon(28)## 9108:

the Heptagonal Pyramidal Number HepPyr(22)## 9126:

the Pentagonal Pyramidal Number PPyr(26), also the 30-gonal Number 30gon(26)## 9131:

the 38-gonal Number 38gon(23)## 9132:

the 35-gonal Number 35gon(24)## 9139:

the Tetrahedral Number Tet(37)## 9153:

the 28-gonal Number 28gon(27), also the Heterogeneous Jonathan Number 28gon(C(3))## 9165:

the 23-gonal Number 23gon(30)## 9177:

the 15-gonal Number 15gon(38)## 9180:

the Triangular Number T(135), also the Hexagonal Number H(68)## 9201:

the Truncated Octahedral Number TruOct(9)## 9211:

the Heptagonal Number Hep(61)## 9216:

a Square, S(96)## 9224:

Octahedral Number Octh(24)## 9261:

the Cube C(21)## 9262:

the 42-gonal Number 42gon(22)## 9296:

Octagonal Number O(56)## 9316:

the Triangular Number T(136), hence the Triangular Jonathan Number T(T(16))## 9322:

a Pentagonal Number P(79)## 9325:

the 33-gonal Number 33gon(25), also the Heterogeneous Jonathan Number 33gon(S(5))## 9331:

the 22-gonal Number 22gon(31), last 2 digits same as index (base 10)## 9334:

the Nonagonal Number N(52)## 9361:

the 16-gonal Number 16gon(37), also the Hendecagonal Number, or 11-gonal Number 11gon(46)## 9367:

the 25-gonal number, 25gon(29)## 9384:

the 39-gonal Number 39gon(23)## 9400:

the 14-gonal Number 14gon(40)## 9408:

the 36-gonal Number 36gon(24)## 9409:

a Square, S(97)## 9433:

the Hexagonal Number H(69)## 9451:

the 31-gonal Number 31gon(26)## 9453:

the Triangular Number T(137)## 9455:

the Square Pyramidal Number SPyr(30)## 9456:

the 21-gonal Number 21gon(32)## 9457:

the Decagonal Number D(49), also the Heterogeneous Jonathan Number D(S(7))## 9478:

the 27-gonal Number, 27gon(28)## 9486:

the 17-gonal Number 17gon(36), also the Heterogeneous Jonathan Number 17gon(S(6))## 9493:

the 43-gonal Number, 43gon(22)## 9500:

the Hexagonal Pyramidal Number HPyr(24)## 9504:

the Dodecagonal Number Dod(44), also the 29-gonal Number 29gon(27), also the Heterogeneous Jonathan Number 29gon(C(3))## 9513:

the 13-gonal Number 13gon(42)## 9517:

the Heptagonal Number Hep(62)## 9537:

the 20-gonal Number 20gon(33)## 9555:

the 18-gonal Number 18gon(35)## 9560:

a Pentagonal Number P(80)## 9571:

the 19-gonal Number 19gon(34)## 9591:

the Triangular Number T(138)## 9600:

the 24-gonal Number 24gon(30)## 9604:

a Square, S(98)## 9625:

the 34-gonal Number 34gon(25), also the Heterogeneous Jonathan Number 34gon(S(5))## 9633:

Octagonal Number O(57)## 9637:

the 40-gonal Number 40gon(23)## 9660:

the Truncated Tetrahedral Number TruTet(14), also the 25-gonal Number 25gon(30)## 9724:

the 44-gonal Number 44gon(22)## 9752:

the 22-gonal Number 22gon(32)## 9672:

the 15-gonal Number 15gon(39)## 9684:

the 37-gonal Number 37gon(24)## 9699:

the Nonagonal Number N(53)## 9730:

the Triangular Number T(139), also the Hexagonal Number H(70)## 9773:

the 26-gonal Number, 26gon(29)## 9776:

the Hendecagonal Number, or 11-gonal Number 11gon(47), also the 32-gonal Number 32gon(26)## 9796:

the 23-gonal Number 23gon(31)## 9801:

a Pentagonal Number P(81), also the Heterogeneous Jonathan Number P(S(9)), and 9 times its reverse, also a Square, S(99)## 9809:

the Stella Octangula Number StOct(17)## 9828:

the Heptagonal Number Hep(63)## 9850:

the Decagonal Number D(50) {to be done: append additional Decagonal Numbers to this sequence}## 9855:

Rhombic Dodecahedral Number RhoDod(14), also the 30-gonal Number 30gon(27), also the Heterogeneous Jonathan Number 30gon(C(3))## 9856:

the 28-gonal Number 28gon(28), also the Diagonal Jonathan Polygonal Number JP(28) = Polygonal(28,28)## 9870:

the Triangular Number T(140)## 9880:

the Tetrahedral Number Tet(38), also the 16-gonal Number 16gon(38)## 9881:

the 14-gonal Number 14gon(41)## 9890:

the 41-gonal Number 41gon(23)## 9891:

the 49-gonal Number 49gon(21)## 9900:

the 54-gonal Number 54gon(20)## 9925:

the 35-gonal Number 35gon(25), also the Heterogeneous Jonathan Number 35gon(S(5))## 9933:

the 111-gonal Number 111gon(14)## 9937:

the 60-gonal Number 60gon(19)## 9945:

the Dodecagonal Number Dod(45), also the 75-gonal Number 75gon(17)## 9952:

the 22-gonal Number 22gon(32)## 9955:

the 45-gonal Number 45gon(22)## 9960:

the 38-gonal Number 38gon(24)## 9963:

the 67-gonal Number 67gon(18)## 9966:

the 183-gonal Number 183gon(11)## 9976:

Octagonal Number O(58), also the 13-gonal Number 13gon(43), also the 358-gonal Number 358gon(8), also the 85-gonal Number 85gon(16), also the Heterogeneous Jonathan Number 85gon(S(4))## 9978:

the 153-gonal Number 153gon(12)## 9981:

the 279-gonal Number 279gon(8)## 9982:

the 477-gonal Number 477gon(7)## 9990:

the 97-gonal Number 97gon(15)## 9995:

the 1001-gonal Number 1001gon(5)## 9996:

the 668-gonal Number 668gon(6)## 9997:

the 130-gonal Number 130gon(13)## 9999:

the Triangular Number T(3334)## 10000:

a Square, S(100), but since 100 is itself a Square, this is the Biquadratic 10^4; also the 224-gonal Number 224gon(10), also the 1668-gonal Number 1668gon(4)## 10001:

10001 = 73 x 137 = 65^2 + 76^2.## 10011:

the Triangular Number T(141)## 10027:

the 17-gonal Number 17gon(37)## 10035:

the 25-gonal number, 25gon(30)## 10045:

a Pentagonal Number P(82)## 10065:

the 21-gonal Number 21gon(33)## 10071:

the Nonagonal Number N(54)## 10101:

the 33-gonal Number 33gon(26), Palindrome alternating two digits (base 10)## 10116:

the 18-gonal Number 18gon(36), also the Heterogeneous Jonathan Number 18gon(S(6))## 10132:

the 20-gonal Number 20gon(34)## 10143:

the 42-gonal Number 42gon(23)## 10144:

the Heptagonal Number Hep(64), also the Heterogeneous Jonathan Number Hep(S(8))## 10150:

the 19-gonal Number 19gon(35)## 10153:

the Triangular Number T(142)## 10179:

the 27-gonal Number, 27gon(29)## 10180:

the 15-gonal Number 15gon(40)## 10200:

the Hendecagonal Number, or 11-gonal Number 11gon(48)## 10201:

a Square, S(101)## 10206:

the Pentagonal Pyramidal Number PPyr(27), also the 31-gonal Number 31gon(27), also the Heterogeneous Jonathan Number 31gon(C(3))## 10225:

the 36-gonal Number 36gon(25), also the Heterogeneous Jonathan Number 36gon(S(5))## 10234:

the 29-gonal Number 29gon(28)## 10236:

the 39-gonal Number 39gon(24)## 10240:

a Constructable Number because (2^11)x5 (power of two times product of Fermat Primes)## 10261:

the 24-gonal Number 24gon(31)## 10280:

a Constructable Number because (2^3)x5x257, a power of two times a product of Fermat Primes## 10292:

a Pentagonal Number P(83)## 10296:

the Triangular Number T(143)## 10325:

Octagonal Number O(59)## 10374:

the 14-gonal Number 14gon(42)## 10396:

the 43-gonal Number, 43gon(23), also the Heptagonal Pyramidal Number HepPyr(23)## 10404:

a Square, S(102)## 10413:

the 16-gonal Number 16gon(39)## 10416:

the Square Pyramidal Number SPyr(31)## 10425:

Octahedral Number Octh(25)## 10426:

the 34-gonal Number 34gon(26), number ends in 2-digit index (base 10)## 10440:

the Triangular Number T(144), also the Heterogeneous Jonathan Number T(S(12))## 10448:

the 23-gonal Number 23gon(32)## 10450:

the Nonagonal Number N(55), also the 13-gonal Number 13gon(44)## 10465:

the Heptagonal Number Hep(65), the last two digits equal the index (base 10)## 10470:

the 26-gonal Number, 26gon(30)## 10512:

the 40-gonal Number 40gon(24)## 10525:

the 37-gonal Number 37gon(25), also the Heterogeneous Jonathan Number 37gon(S(5))## 10542:

a Pentagonal Number P(84)## 10557:

the 32-gonal Number 32gon(27), also the Heterogeneous Jonathan Number 32gon(C(3))## 10583:

the 17-gonal Number 17gon(38)## 10585:

the Triangular Number T(145), also the 28-gonal Number 28gon(29)## 10593:

the 22-gonal Number 22gon(33)## 10609:

a Square, S(103)## 10612:

the 30-gonal Number 30gon(28)## 10626:

the Pentatope Number Ptop(21)## 10633:

the Hendecagonal Number, or 11-gonal Number 11gon(49), also the Heterogeneous Jonathan Number 11-gon(S(7))## 10648:

the Cube C(22)## 10649:

the 44-gonal Number 44gon(23)## 10660:

the Tetrahedral Number Tet(39)## 10680:

Octagonal Number O(60)## 10693:

the 18-gonal Number 18gon(37), also the 21-gonal Number 21gon(34)## 10701:

the 15-gonal Number 15gon(41)## 10725:

the Hexagonal Pyramidal Number HPyr(25)## 10731:

the Triangular Number T(146)## 10745:

a Centered Cube Number CC(18), also the 20-gonal Number 20gon(35)## 10746:

the 19-gonal Number 19gon(36), also the Heterogeneous Jonathan Number 19gon(S(6))## 10751:

the 35-gonal Number 35gon(26)## 10788:

the 41-gonal Number 41gon(24)## 10791:

the Heptagonal Number Hep(66)## 10795:

a Pentagonal Number P(85)## 10816:

a Square, S(104)## 10825:

the 38-gonal Number 38gon(25), also the Heterogeneous Jonathan Number 38gon(S(5))## 10836:

the Nonagonal Number N(56)## 10878:

the Triangular Number T(147)## 10879:

the 14-gonal Number 14gon(43)## 10880:

a Constructable Number because (2^7)x5x17, a power of two times a product of Fermat Primes## 10905:

the 27-gonal Number, 27gon(30) {to be done: append additional 27gonal Numbers in this sequence)## 10908:

the 33-gonal Number 33gon(27), also the Heterogeneous Jonathan Number 33gon(C(3))## 10935:

the 13-gonal Number 13gon(45)## 10944:

the 24-gonal Number 24gon(32)## 10960:

the 16-gonal Number 16gon(40) {to be done: append additional 16gonal Numbers in this sequence}## 10990:

the 31-gonal Number 31gon(28)## 10991:

the 29-gonal Number 29gon(29)## 11025:

a Square, S(105), also the Heterogeneous Jonathan Number S(T(14))## 11026:

the Triangular Number T(148)## 11041:

Octagonal Number O(61)## 11051:

a Pentagonal Number P(86)## 11064:

the 42-gonal Number 42gon(24)## 11075:

the Hendecagonal Number, or 11-gonal Number 11gon(50) {to be done: append additional 11-gonal Numbers in this sequence}## 11076:

the 36-gonal Number 36gon(26)## 11121:

the 23-gonal Number 23gon(33)## 11122:

the Heptagonal Number Hep(67), note five digits in ascending order, also as a Poker hand: a Full House of Aces over Deuces## 11125:

the 39-gonal Number 39gon(25), also the Heterogeneous Jonathan Number 39gon(S(5))## 11154:

the 17-gonal Number 17gon(39)## 11616:

the 44-gonal Number 44gon(24)## 11175:

the Triangular Number T(149)## 11229:

the Nonagonal Number N(57)## 11235:

the 15-gonal Number 15gon(42), also the 15gonal Jonathan Number 15gon(15gon(3))## 11236:

a Square, S(106)## 11254:

the 22-gonal Number 22gon(34)## 11259:

the 34-gonal Number 34gon(27), also the Heterogeneous Jonathan Number 34gon(C(3))## 11286:

the 18-gonal Number 18gon(38)## 11310:

a Pentagonal Number P(87), also the product of two Pentagonals: P(2)P(39)## 11325:

the Triangular Number T(150)## 11340:

the 28-gonal Number 28gon(30) {to be done: append additional Nonagonal numbers in this sequence}, also the 21-gonal Number 21gon(35), also the 43-gonal Number, 43gon(24)## 11359:

the 19-gonal Number 19gon(37)## 11368:

the Pentagonal Pyramidal Number PPyr(28), also the 32-gonal Number 32gon(28)## 11376:

the 20-gonal Number 20gon(36), also the Heterogeneous Jonathan Number 20gon(S(6))## 11396:

the 14-gonal Number 14gon(44)## 11397:

the 30-gonal Number 30gon(29)## 11401:

the 37-gonal Number 37gon(26)## 11408:

Octagonal Number O(62)## 11425:

the 40-gonal Number 40gon(25), also the Heterogeneous Jonathan Number 40gon(S(5))## 11431:

the 13-gonal Number 13gon(46)## 11440:

the Square Pyramidal Number SPyr(32)## 11449:

a Square, S(107)## 11476:

the Triangular Number T(151)## 11480:

the Tetrahedral Number Tet(40)## 11458:

the Heptagonal Number Hep(68)## 11572:

a Pentagonal Number P(88)## 11610:

the 35-gonal Number 35gon(27), also the Heterogeneous Jonathan Number 35gon(C(3))## 11628:

the Triangular Number T(152)## 11629:

the Nonagonal Number N(58)## 11646:

the Stella Octangula Number StOct(18)## 11649:

the 24-gonal Number 24gon(33)## 11664:

a Square, S(108)## 11725:

the 41-gonal Number 41gon(25), also the Heterogeneous Jonathan Number 41gon(S(5))## 11726:

Octahedral Number Octh(26), also the 38-gonal Number 38gon(26)## 11740:

the 17-gonal Number 17gon(40) {to be done: append additional 17gon Numbers in this sequence}## 11746:

the 33-gonal Number 33gon(28)## 11775:

the 29-gonal Number 29gon(30) {to be done: append additional 29gon Numbers in this sequence}## 11781:

the Triangular Number T(153), also the the Triangular Jonathan Number T(T(17)), also the Hexagonal Number H(__), also Octagonal Number O(63)## 11782:

the 15-gonal Number 15gon(43)## 11799:

the Heptagonal Number Hep(69)## 11800:

the Heptagonal Pyramidal Number HepPyr(24)## 11803:

the 31-gonal Number 31gon(29)## 11815:

the 23-gonal Number 23gon(34)## 11837:

a Pentagonal Number P(89)## 11881:

a Square, S(109)## 11895:

the 18-gonal Number 18gon(39)## 11925:

the 14-gonal Number 14gon(45)## 11935:

the Triangular Number T(154), also the 22-gonal Number 22gon(35), last 2 digits same as index (base 10)## 11938:

the 13-gonal Number 13gon(47)## 11950:

the Truncated Tetrahedral Number TruTet(15)## 11961:

the 36-gonal Number 36gon(27), also the Heterogeneous Jonathan Number 36gon(C(3))## 11989:

the 19-gonal Number 19gon(38)## 12006:

the 21-gonal Number 21gon(36), also the Heterogeneous Jonathan Number 21gon(S(6))## 12025:

the 20-gonal Number 20gon(37), also the 42-gonal Number 42gon(25), also the Heterogeneous Jonathan Number 42gon(S(5))## 12036:

the Nonagonal Number N(59)## 12051:

the Hexagonal Pyramidal Number HPyr(26); also the 39-gonal Number 39gon(26)## 12090:

the Triangular Number T(155)## 12100:

a Square, S(110)## 12105:

a Pentagonal Number P(90)## 12124:

the 34-gonal Number 34gon(28)## 12145:

the Heptagonal Number Hep(70)## 12160:

Octagonal Number O(64), also the Heterogeneous Jonathan Number O(S(8))## 12167:

the Cube C(23)## 12209:

the 32-gonal Number 32gon(29), also Rhombic Dodecahedral Number RhoDod(15), Rhombic Dodecahedral Jonathan Number RhoDod(RhoDod(2))## 12210:

the 30-gonal Number 30gon(30) {to be done: append additional 30gon Numbers in this sequence}## 12325:

the 43-gonal Number, 43gon(25), also the Heterogeneous Jonathan Number 43gon(S(5))## 12376:

the 40-gonal Number 40gon(26)## 12246:

the Triangular Number T(156)## 12288:

a Constructable Number because (2^12)x3 (power of two times product of Fermat Primes)## 12312:

the 37-gonal Number 37gon(27), also the Heterogeneous Jonathan Number 37gon(C(3))## 12336:

a Constructable Number because (2^4)x3x257, a power of two times a product of Fermat Primes## 12321:

a Square, S(111), also a Palindrome## 12341:

the Tetrahedral Number Tet(41)## 12342:

the 15-gonal Number 15gon(44)## 12376:

a Pentagonal Number P(91), also the Decagonal Number D(56), also the 24-gonal Number 24gon(34)## 12403:

the Triangular Number T(157)## 12450:

the Nonagonal Number N(60)## 12456:

the 13-gonal Number 13gon(48)## 12466:

the 14-gonal Number 14gon(46)## 12496:

the Heptagonal Number Hep(71)## 12502:

the 35-gonal Number 35gon(28)## 12520:

the 18-gonal Number 18gon(40) {to be done: append additional 18gon Numbers in this sequence}## 12529:

the Square Pyramidal Number SPyr(33)## 12530:

the 23-gonal Number 23gon(35)## 12544:

a Square, S(112)## 12545:

Octagonal Number O(65), also Octagonal Jonathan Number O(O(5))## 12561:

the Triangular Number T(158)## 12615:

the 33-gonal Number 33gon(29), also the Pentagonal Pyramidal Number PPyr(29)## 12625:

the 44-gonal Number 44gon(25), also the Heterogeneous Jonathan Number 44gon(S(5))## 12636:

the 19-gonal Number 19gon(39), also the 22-gonal Number 22gon(36), also the Heterogeneous Jonathan Number 22gon(S(6))## 12645:

the 31-gonal Number 31gon(30)## 12650:

the Pentatope Number Ptop(22), also a Pentagonal Number P(92)## 12663:

the 38-gonal Number 38gon(27), also the Heterogeneous Jonathan Number 38gon(C(3))## 12691:

a Centered Cube Number CC(19), also the 21-gonal Number 21gon(37)## 12692:

the 20-gonal Number 20gon(38)## 12701:

the 41-gonal Number 41gon(26)## 12720:

the Triangular Number T(159)## 12769:

a Square, S(113)## 12852:

the Heptagonal Number Hep(72)## 12871:

the Nonagonal Number N(61)## 12880:

the Triangular Number T(160), also the 36-gonal Number 36gon(28)## 12915:

the 15-gonal Number 15gon(45)## 12927:

a Pentagonal Number P(93)## 12934:

the Truncated Octahedral Number TruOct(10)## 12936:

Octagonal Number O(66)## 12985:

the 13-gonal Number 13gon(49), also the Heterogeneous Jonathan Number 13gon(S(7))## 12996:

a Square, S(114)## 13014:

the 39-gonal Number 39gon(27), also the Heterogeneous Jonathan Number 39gon(C(3))## 13019:

the 14-gonal Number 14gon(47)## 13021:

the 34-gonal Number 34gon(29)## 13041:

the Triangular Number T(161)## 13056:

a Constructable Number because (2^8)x3x17, a power of two times a product of Fermat Primes## 13080:

the 32-gonal Number 32gon(30)## 13107:

a Constructable Number because 3x17x257, a power of two (here 2^0=1) times a product of Fermat Primes## 13125:

the 24-gonal Number 24gon(35)## 13131:

Octahedral Number Octh(27), also a Palindrome## 13203:

the Triangular Number T(162)## 13207:

a Pentagonal Number P(94)## 13213:

the Heptagonal Number Hep(73)## 13225:

a Square, S(115)## 13026:

the 42-gonal Number 42gon(26), number ends in 2-digit index (base 10)## 13244:

the Tetrahedral Number Tet(42)## 13258:

the 37-gonal Number 37gon(28)## 13266:

the 23-gonal Number 23gon(36), also the Heterogeneous Jonathan Number 23gon(S(6))## 13299:

the Nonagonal Number N(62)## 13300:

the 19-gonal Number 19gon(40) {to be done: append additional 19-gonal Numbers in this sequence}## 13325:

the Heptagonal Pyramidal Number HepPyr(25)## 13333:

Octagonal Number O(67)## 13351:

the 43-gonal Number, 43gon(26)## 13357:

the 22-gonal Number 22gon(37)## 13365:

the 40-gonal Number 40gon(27), also the Heterogeneous Jonathan Number 40gon(C(3))## 13366:

the Triangular Number T(163)## 13377:

the 20-gonal Number 20gon(39)## 13395:

the 21-gonal Number 21gon(38)## 13427:

the 35-gonal Number 35gon(29)## 13456:

a Square, S(116)## 13482:

the Hexagonal Pyramidal Number HPyr(27)## 13490:

a Pentagonal Number P(95)## 13501:

the 15-gonal Number 15gon(46)## 13515:

the 33-gonal Number 33gon(30)## 13516:

the 31-gonal Number 31gon(31) {to be done: append additional 31-gonal Numbers in this sequence} also the 31-gonal Jonathan Number 31gon(31gon(2))## 13525:

the 13-gonal Number 13gon(50)## 13530:

the Triangular Number T(164)## 13579:

the Heptagonal Number Hep(74), note this is all the odd digits in ascending order (base 10)## 13584:

the 14-gonal Number 14gon(48)## 13636:

the 38-gonal Number 38gon(28)## 13676:

the 44-gonal Number 44gon(26)## 13685:

the Square Pyramidal Number SPyr(34)## 13689:

a Square, S(117)## 13695:

the Triangular Number T(165)## 13699:

the Stella Octangula Number StOct(19)## 13716:

the 41-gonal Number 41gon(27), also the Heterogeneous Jonathan Number 41gon(C(3))## 13734:

the Nonagonal Number N(63)## 13736:

Octagonal Number O(68)## 13776:

a Pentagonal Number P(96)## 13824:

the Cube C(24)## 13833:

the 36-gonal Number 36gon(29)## 13861:

the Triangular Number T(166)## 13896:

the 24-gonal Number 24gon(36), also the Heterogeneous Jonathan Number 24gon(S(6))## 13924:

a Square, S(118)## 13950:

the Pentagonal Pyramidal Number PPyr(30), also the Heptagonal Number Hep(75), also the 34-gonal Number 34gon(30) {to be done: append additional 34-gonal Numbers in this sequence}## 13981:

the 32-gonal Number 32gon(31)## 14014:

the 39-gonal Number 39gon(28), number starts and ends with 2-gits equal to half of index (base 10)## 14023:

the 23-gonal Number 23gon(37)## 14028:

the Triangular Number T(167)## 14065:

a Pentagonal Number P(97)## 14067:

the 42-gonal Number 42gon(27), also the Heterogeneous Jonathan Number 42gon(C(3))## 14080:

the 20-gonal Number 20gon(40) {to be done: append additional 20-gonal Numbers to this sequence}## 14098:

the 22-gonal Number 22gon(38)## 14100:

the 15-gonal Number 15gon(47)## 14118:

the 21-gonal Number 21gon(39)## 14145:

Octagonal Number O(69)## 14161:

a Square, S(119), also the 14-gonal Number 14gon(49), also the Heterogeneous Jonathan Number 14gon(S(7))## 14176:

the Nonagonal Number N(64), also the Heterogeneous Jonathan Number N(S(8))## 14190:

the Tetrahedral Number Tet(43)## 14196:

the Triangular Number T(168)## 14239:

the 37-gonal Number 37gon(29)## 14326:

the Heptagonal Number Hep(76)## 14357:

a Pentagonal Number P(98)## 14365:

the Triangular Number T(169) also the Heterogeneous Jonathan Number T(S(13))## 14385:

the 35-gonal Number 35gon(30) {to be done: append additional 35-gonal Numbers to this sequence}## 14392:

the 40-gonal Number 40gon(28)## 14400:

a Square, S(120), also the Heterogeneous Jonathan Number S(T(15)) also the Heterogeneous Jonathan Number S(T(T(5)))## 14418:

the 43-gonal Number, 43gon(27), also the Heterogeneous Jonathan Number 43gon(C(3))## 14446:

the 33-gonal Number 33gon(31)## 14535:

the Triangular Number T(170)## 14560:

Octagonal Number O(70)## 14576:

the Truncated Tetrahedral Number TruTet(16), also the Truncated Tetrahedral Jonathan Number TruTet(TruTet(2))## 14625:

the Nonagonal Number N(65)## 14641:

a Square, S(121), but since 121 is itself a Square, this is the Biquadratic 11^4## 14644:

Octahedral Number Octh(28)## 14645:

the 38-gonal Number 38gon(29)## 14652:

a Pentagonal Number P(99)## 14689:

the 24-gonal Number 24gon(37)## 14706:

the Triangular Number T(171), also the Jonathan Number T(T(18))## 14707:

the Heptagonal Number Hep(77)## 14712:

the 15-gonal Number 15gon(48)## 14750:

the 14-gonal Number 14gon(50) {to be done: append additional 14-gonal numbers in this sequence}## 14769:

the 44-gonal Number 44gon(27), also the Heterogeneous Jonathan Number 44gon(C(3))## 14770:

the 41-gonal Number 41gon(28)## 14801:

the 23-gonal Number 23gon(38)## 14820:

the 36-gonal Number 36gon(30) {to be done: append additional 36-gonal numbers in this sequence}## 14859:

a Centered Cube Number CC(20), also the 22-gonal Number 22gon(39)## 14860:

the 21-gonal Number 21gon(40) {to be done: append additional 21-gonal numbers in this sequence}## 14878:

the Triangular Number T(172)## 14884:

a Square, S(122)## 14910:

the Square Pyramidal Number SPyr(35)## 14911:

Rhombic Dodecahedral Number RhoDod(16)## 14912:

the 32-gonal Number 32gon(32), {to be done: append additional 32-gonal Numbers to this sequence} also the 32-gonal Jonathan Number 32gon(32gon(2))## 14950:

the Pentatope Number Ptop(23), also the Pentagonal Number P(100), also the Heterogeneous Jonathan Number P(S(10))## 14976:

the Heptagonal Pyramidal Number HepPyr(26), also the Heptagonal Pyramidal Jonathan Number HepPyr(HepPyr(3))## 14981:

Octagonal Number O(71)## 15022:

the Hexagonal Pyramidal Number HPyr(28)## 15051:

the Triangular Number T(173), also the 39-gonal Number 39gon(29); also a Palindrome## 15081:

the Nonagonal Number N(66)## 15093:

the Heptagonal Number Hep(78)## 15129:

a Square, S(123)## 15148:

the 42-gonal Number 42gon(28)## 15180:

the Tetrahedral Number Tet(44)## 15225:

the Triangular Number T(174)## 15251:

the Pentagonal Number P(101), also a palindrome, {to be done: add the Pentagonal Numbers P(102) through P(300) per printout}## 15255:

the 37-gonal Number 37gon(30) {to be done: append additional 37-gonal Numbers in this sequence}## 15337:

the 15-gonal Number 15gon(49), also the Heterogeneous Jonathan Number 15gon(S(7))## 15360:

a Constructable Number because (2^10)x3x5 (power of two times product of Fermat Primes)## 15376:

the Pentagonal Pyramidal Number PPyr(31), also a Square, S(124)## 15400:

the Triangular Number T(175)## 15408:

Octagonal Number O(72), also the 33-gonal Number 33gon(32)## 15420:

a Constructable Number because (2^2)x3x5x257, a power of two times a product of Fermat Primes## 15457:

the 40-gonal Number 40gon(29)## 15484:

the Heptagonal Number Hep(79)## 15504:

the 24-gonal Number 24gon(38)## 15526:

the 43-gonal Number, 43gon(28)## 15544:

the Nonagonal Number N(67)## 15576:

the Triangular Number T(176)## 15600:

the 23-gonal Number 23gon(39)## 15625:

a Square, S(125), also the Cube C(25), since 25 is itself a Square, we have a 6th power, 5^6## 15640:

the 22-gonal Number 22gon(40) {to be done: append additional 22-gonal Numbers in this sequence}## 15690:

the 38-gonal Number 38gon(30) {to be done: append additional 38-gonal Numbers in this sequence}## 15753:

the Triangular Number T(177)## 15841:

Octagonal Number O(73)## 15863:

the 41-gonal Number 41gon(29)## 15876:

a Square, S(126)## 15880:

the Heptagonal Number Hep(80), note that the index equals the last two digits (base 10)## 15904:

the 44-gonal Number 44gon(28)## 15931:

the Triangular Number T(178)## 15975:

the 15-gonal Number 15gon(50) {to be done: append additional 15gon Numbers in this sequence}## 15980:

the Stella Octangula Number StOct(20)## 16014:

the Nonagonal Number N(68)## 16110:

the Triangular Number T(179)## 16125:

the 39-gonal Number 39gon(30) {to be done: append additional 39-gonal Numbers in this sequence}## 16129:

a Square, S(127)## 16206:

the Square Pyramidal Number SPyr(36)## 16269:

Octahedral Number Octh(29), also the 42-gonal Number 42gon(29)## 16280:

Octagonal Number O(74)## 16281:

the Heptagonal Number Hep(81) note that the index equals the last two digits of the Heptagon, base 10), also the Heterogeneous Jonathan Number Hep(S(9)), also the Heptagonal Jonathan Number Hep(Hep(6))## 16290:

the Triangular Number T(180)## 16320:

a Constructable Number because (2^6)x3x5x17, a power of two times a product of Fermat Primes## 16341:

the 24-gonal Number 24gon(39)## 16384:

a Square, S(128), but since 128 is a 7th power, we have a 14th power 2^14## 16401:

the 33-gonal Number 33gon(33), also the 33-gonal Jonathan Number 33gon(33gon(2))## 16420:

the 23-gonal Number 23gon(40)## 16448:

a Constructable Number because (2^6)x257, a power of two times a product of Fermat Primes## 16471:

the Triangular Number T(181)## 16491:

the Nonagonal Number N(69)## 16560:

the 40-gonal Number 40gon(30) {to be done: append additional 40-gonal Numbers in this sequence}## 16641:

a Square, S(129)## 16653:

the Triangular Number T(182)## 16675:

the 43-gonal Number, 43gon(29), also the Hexagonal Pyramidal Number HPyr(29)## 16687:

the Heptagonal Number Hep(82)## 16725:

Octagonal Number O(75)## 16758:

the Heptagonal Pyramidal Number HepPyr(27)## 16836:

the Triangular Number T(183)## 16896:

the Pentagonal Pyramidal Number PPyr(32)## 16900:

a Square, S(130)## 16975:

the Nonagonal Number N(70)## 16995:

the 41-gonal Number 41gon(30) {to be done: append additional 41-gonal Numbers in this sequence}## 17020:

the Triangular Number T(184)## 17081:

the 44-gonal Number 44gon(29)## 17098:

the Heptagonal Number Hep(83)## 17161:

a Square, S(131)## 17176:

Octagonal Number O(76), last two digits of index equal last two digits of Octagonal (Base 10)## 17200:

the 24-gonal Number 24gon(40) {to be done: append additional 24-gonal Numbers in this sequence}## 17205:

the Triangular Number T(185)## 17261:

a Centered Cube Number CC(21)## 17391:

the Triangular Number T(186)## 17408:

a Constructable Number because (2^10)x17 (power of two times product of Fermat Primes)## 17424:

a Square, S(132)## 17430:

the 42-gonal Number 42gon(30) {to be done: append additional 42-gonal Numbers in this sequence}## 17466:

the Nonagonal Number N(71)## 17476:

a Constructable Number because (2^2)x17x257, a power of two times a product of Fermat Primes## 17514:

the Heptagonal Number Hep(84)## 17550:

the Pentatope Number Ptop(24)## 17561:

the Truncated Octahedral Number TruOct(11), and also the Truncated Tetrahedral Number TruTet(17); this is the first nontrivial Truncated Tetrahedral Truncated Octahedral Number## 17575:

the Square Pyramidal Number SPyr(37)## 17576:

the Cube C(26)## 17578:

the Triangular Number T(187)## 17633:

Octagonal Number O(77)## 17689:

a Square, S(133)## 17766:

the Triangular Number T(188)## 17865:

the 43-gonal Number, 43gon(30) {to be done: append additional 43-gonal Numbers in this sequence}## 17935:

the Heptagonal Number Hep(85)## 17955:

the Triangular Number T(189)## 17956:

a Square, S(134)## 17964:

the Nonagonal Number N(72)## 17985:

Rhombic Dodecahedral Number RhoDod(17)## 18010:

Octahedral Number Octh(30)## 18096:

Octagonal Number O(78)## 18145:

the Triangular Number T(190), also the Triangular Jonathan Number T(T(19))## 18225:

a Square, S(135)## 18300:

the 44-gonal Number 44gon(30) {to be done: append additional 44-gonal Numbers in this sequence}## 18336:

the Triangular Number T(191)## 18361:

the Heptagonal Number Hep(86)## 18445:

the Hexagonal Pyramidal Number HPyr(30)## 18469:

the Nonagonal Number N(73)## 18496:

a Square, S(136), also the Heterogeneous Jonathan Number S(T(16)) also the Heterogeneous Jonathan Number S(T(S(4)))## 18501:

the Stella Octangula Number StOct(21)## 18513:

the Pentagonal Pyramidal Number PPyr(33)## 18528:

the Triangular Number T(192)## 18565:

Octagonal Number O(79)## 18676:

the Heptagonal Pyramidal Number HepPyr(28)## 18721:

the Triangular Number T(193)## 18769:

a Square, S(137)## 18792:

the Heptagonal Number Hep(87)## 18915:

the Triangular Number T(194)## 18981:

the Nonagonal Number N(74)## 19019:

the Square Pyramidal Number SPyr(38)## 19040:

Octagonal Number O(80)## 19044:

a Square, S(138)## 19110:

the Triangular Number T(195)## 19228:

the Heptagonal Number Hep(88)## 19306:

the Triangular Number T(196)## 19321:

a Square, S(139)## 19500:

the Nonagonal Number N(75), also the Nonagonal Jonathan Number N(N(5))## 19503:

the Triangular Number T(197)## 19521:

Octagonal Number O(81), also Heterogeneous Jonathan Number O(S(8))## 19600:

the largest of the three numbers to be both Tetrahedral and Square, Tet(48) = S(140), the others being the Tet(1)=S(1) and Tet(2)=S(2), as proven by Meyl (1878, cited by Dickson, 1952, p.25); also the Square, S(140)## 19669:

the Heptagonal Number Hep(89)## 19683:

the Cube C(27) since 27 is itself a Cube, we have a 9th power, 3^9## 19701:

the Triangular Number T(198)## 19871:

Octahedral Number Octh(31)## 19881:

a Square, S(141)## 19900:

the Triangular Number T(199)## 19909:

a Centered Cube Number CC(21)## 20008:

Octagonal Number O(82)## 20026:

the Nonagonal Number N(76)## 20100:

the Triangular Number T(200)## 20115:

the Heptagonal Number Hep(90)## 20164:

a Square, S(142)## 20230:

the Pentagonal Pyramidal Number PPyr(34)## 20301:

the Triangular Number T(201)## 20336:

the Hexagonal Pyramidal Number HPyr(31)## 20449:

a Square, S(143)## 20475:

the Pentatope Number Ptop(25)## 20480:

a Constructable Number because (2^12)x5 (power of two times product of Fermat Primes)## 20501:

Octagonal Number O(83)## 20503:

the Triangular Number T(202)## 20540:

the Square Pyramidal Number SPyr(39)## 20559:

the Nonagonal Number N(77)## 20560:

a Constructable Number because (2^4)x5x257, a power of two times a product of Fermat Primes## 20566:

the Heptagonal Number Hep(91)## 20706:

the Triangular Number T(203)## 20735:

the Heptagonal Pyramidal Number HepPyr(29)## 20736:

a Square, S(144), but since 144 is a Sqaure, we have the Biquadratic 12^4## 20910:

the Triangular Number T(204)## 20928:

Truncated Tetrahedral Number TruTet(18)## 21000:

Octagonal Number O(84)## 21022:

the Heptagonal Number Hep(92)## 21025:

a Square, S(145)## 21099:

the Nonagonal Number N(78)## 21115:

the Triangular Number T(205)## 21274:

the Stella Octangula Number StOct(22)## 21316:

a Square, S(146)## 21321:

the Triangular Number T(206)## 21455:

Rhombic Dodecahedral Number RhoDod(18)## 21483:

the Heptagonal Number Hep(93)## 21505:

Octagonal Number O(85)## 21528:

the Triangular Number T(207)## 21609:

a Square, S(147)## 21736:

the Triangular Number T(208)## 21646:

the Nonagonal Number N(79)## 21760:

a Constructable Number because (2^8)x5x17, a power of two times a product of Fermat Primes## 21845:

a Constructable Number because 5x17x257, a power of two (here 2^0=1) times a product of Fermat Primes## 21856:

Octahedral Number Octh(32)## 21904:

a Square, S(148)## 21945:

the Triangular Number T(209)## 21949:

the Heptagonal Number Hep(94)## 21952:

the Cube C(28)## 22016:

Octagonal Number O(86)## 22050:

the Pentagonal Pyramidal Number PPyr(35)## 22140:

the Square Pyramidal Number SPyr(40)## 22155:

the Triangular Number T(210), also the Triangular Jonathan Number T(T(20))## 22200:

the Nonagonal Number N(80)## 22201:

a Square, S(149)## 22352:

the Hexagonal Pyramidal Number HPyr(32)## 22366:

the Triangular Number T(211)## 22420:

the Heptagonal Number Hep(95)## 22500:

a Square, S(150)## 22533:

Octagonal Number O(87)## 22578:

the Triangular Number T(212)## 22761:

the Nonagonal Number N(81), also the Heterogeneous Jonathan Number N(S(9))## 22791:

the Triangular Number T(213)## 22801:

a Square, S(151)## 22896:

the Heptagonal Number Hep(96), note index equals last two digits of heptagonal (base 10)## 22940:

the Heptagonal Pyramidal Number HepPyr(30)## 23005:

the Triangular Number T(214)## 23056:

Octagonal Number O(88)## 23104:

a Square, S(152)## 23178:

the Truncated Octahedral Number TruOct(12)## 23220:

the Triangular Number T(215)## 23329:

the Nonagonal Number N(82)## 23377:

the Heptagonal Number Hep(97)## 23409:

a Square, S(153), also the Heterogeneous Jonathan Number S(T(17))## 23436:

the Triangular Number T(216)## 23585:

Octagonal Number O(89)## 23653:

the Triangular Number T(217)## 23716:

a Square, S(154)## 23751:

the Pentatope Number Ptop(26)## 23821:

the Square Pyramidal Number SPyr(41)## 23863:

the Heptagonal Number Hep(98)## 23871:

the Triangular Number T(218)## 23904:

the Nonagonal Number N(83)## 23969:

Octahedral Number Octh(33)## 23976:

the Pentagonal Pyramidal Number PPyr(36)## 24025:

a Square, S(155)## 24090:

the Triangular Number T(219)## 24120:

Octagonal Number O(90), note the first two digits 24=4! and last 3 digits 120=5!## 24310:

the Triangular Number T(220)## 24311:

the Stella Octangula Number StOct(23)## 24336:

a Square, S(156)## 24354:

the Heptagonal Number Hep(99)## 24389:

the Cube C(29)## 24486:

the Nonagonal Number N(84)## 24497:

the Hexagonal Pyramidal Number HPyr(33)## 24531:

the Triangular Number T(221)## 24576:

a Constructable Number because (2^13)x3 (power of two times product of Fermat Primes)## 24649:

a Square, S(157)## 24661:

Octagonal Number O(91)## 24672:

a Constructable Number because (2^5)x3x257, a power of two times a product of Fermat Primes## 24700:

Truncated Tetrahedral Number TruTet(19)## 24753:

the Triangular Number T(222)## 24850:

the Heptagonal Number Hep(100), also the Heterogeneous Jonathan Number Hep(S(10))## 24952:

a Constructable Number because (2^3)x17x257, a power of two times a product of Fermat Primes## 24964:

a Square, S(158)## 24976:

the Triangular Number T(223)## 25075:

the Nonagonal Number N(85)## 25200:

the Triangular Number T(224)## 25208:

Octagonal Number O(92)## 25281:

a Square, S(159)## 25296:

the Heptagonal Pyramidal Number HepPyr(31)## 25345:

Rhombic Dodecahedral Number RhoDod(19)## 25351:

the Heptagonal Number Hep(101)## 25425:

the Triangular Number T(225), also the Heterogeneous Jonathan Number T(S(15))## 25585:

the Square Pyramidal Number SPyr(42)## 25600:

a Square, S(160)## 25651:

the Triangular Number T(226)## 25671:

the Nonagonal Number N(86)## 25761:

Octagonal Number O(93)## 25857:

the Heptagonal Number Hep(102)## 25878:

the Triangular Number T(227)## 25921:

a Square, S(161)## 26011:

the Pentagonal Pyramidal Number PPyr(37)## 26106:

the Triangular Number T(228)## 26112:

a Constructable Number because (2^9)x3x17, a power of two times a product of Fermat Primes## 26214:

Octahedral Number Octh(34), and a Constructable Number because 2x3x17x257, a power of two times a product of Fermat Primes (I'd call that a coincidence, but there are no coincidences in Mathematics)## 26244:

a Square, S(162)## 26274:

the Nonagonal Number N(87)## 26320:

Octagonal Number O(94)## 26335:

the Triangular Number T(229)## 26368:

the Heptagonal Number Hep(103)## 26565:

the Triangular Number T(230)## 26569:

a Square, S(163)## 26775:

the Hexagonal Pyramidal Number HPyr(34)## 26796:

the Triangular Number T(231), also the the Triangular Jonathan Number T(2,21) = T(T(21)), also the Triangular Jonathan Number T(3,6) = T(T(T(6))), also the Triangular Jonathan Number T(4,3) = T(T(T(T(3)))), also the Triangular Jonathan Number T(5,2) = T(T(T(T(T(2)))))## 26884:

the Heptagonal Number Hep(104), also the Nonagonal Number N(88)## 26885:

Octagonal Number O(95)## 26896:

a Square, S(164)## 27000:

the Cube C(30)## 27028:

the Triangular Number T(232)## 27225:

a Square, S(165)## 27261:

the Triangular Number T(233)## 27405:

the Pentatope Number Ptop(27), also the Heptagonal Number Hep(105)## 27495:

the Triangular Number T(234)## 27434:

the Square Pyramidal Number SPyr(43)## 27456:

Octagonal Number O(96), also Octagonal Jonathan Number O(O(6))## 27501:

the Nonagonal Number N(89)## 27556:

a Square, S(166)## 27624:

the Stella Octangula Number StOct(24)## 27730:

the Triangular Number T(235)## 27808:

the Heptagonal Pyramidal Number HepPyr(32)## 27931:

the Heptagonal Number Hep(106)## 28033:

Octagonal Number O(97)## 28125:

the Nonagonal Number N(90)## 28158:

the Pentagonal Pyramidal Number PPyr(38)## 28595:

Octahedral Number Octh(35)## 27889:

a Square, S(167)## 27931:

the Heptagonal Number Hep(106)## 27966:

the Triangular Number T(236)## 28203:

the Triangular Number T(237)## 28224:

a Square, S(168)## 28441:

the Triangular Number T(238)## 28462:

the Heptagonal Number Hep(107)## 28561:

a Square, S(169), but since 169 is a Square, we have the Biquadratic 13^4## 28616:

Octagonal Number O(98)## 28680:

the Triangular Number T(239)## 28756:

the Nonagonal Number N(91)## 28900:

Truncated Tetrahedral Number TruTet(20), also the Square S(170), this is the third Square Truncated Tetrahedral Number, the others being TruTet(1)=1=1^4, and TruTet(2)=6=2^4,## 28920:

the Triangular Number T(240)## 28998:

the Heptagonal Number Hep(108)## 29161:

the Triangular Number T(241)## 29190:

the Hexagonal Pyramidal Number HPyr(35)## 29205:

Octagonal Number O(99)## 29241:

a Square, S(171), also the Heterogeneous Jonathan Number S(T(18))## 29370:

the Square Pyramidal Number SPyr(44) {to be done: append additional numbers in this sequence}## 29394:

the Nonagonal Number N(92)## 29403:

the Triangular Number T(242)## 29539:

the Heptagonal Number Hep(109)## 29584:

a Square, S(172)## 29646:

the Triangular Number T(243)## 29679:

Rhombic Dodecahedral Number RhoDod(20)## 29791:

the Cube C(31)## 29800:

Octagonal Number O(100)## 29881:

the Truncated Octahedral Number TruOct(13)## 29890:

the Triangular Number T(244)## 29929:

a Square, S(173)## 30039:

the Nonagonal Number N(93)## 30085:

the Heptagonal Number Hep(110)## 30135:

the Triangular Number T(245)## 30276:

a Square, S(174)## 30381:

the Triangular Number T(246)## 30420:

the Pentagonal Pyramidal Number PPyr(39) {to be done: append additional numbers in this sequence}## 30481:

the Heptagonal Pyramidal Number HepPyr(33)## 30625:

a Square, S(175)## 30628:

the Triangular Number T(247)## 30636:

the Heptagonal Number Hep(111)## 30691:

the Nonagonal Number N(94)## 30720:

a Constructable Number because (2^11)x3x5 (power of two times product of Fermat Primes)## 30876:

the Triangular Number T(248)## 30976:

a Square, S(176)## 30840:

a Constructable Number because (2^3)x3x5x257, a power of two times a product of Fermat Primes## 31116:

Octahedral Number Octh(36)## 31125:

the Triangular Number T(249)## 31192:

the Heptagonal Number Hep(112), also the Heptagonal Jonathan Number Hep(Hep(7)), also the Heptagonal Jonathan Number Hep(3,2) = Hep(Hep(Hep(2)))## 31225:

the Stella Octangula Number StOct(25)## 31329:

a Square, S(177)## 31350:

the Nonagonal Number N(95)## 31375:

the Triangular Number T(250)## 31465:

the Pentatope Number Ptop(28)## 31626:

the Triangular Number T(251)## 31684:

a Square, S(178)## 31746:

the Hexagonal Pyramidal Number HPyr(36)## 31753:

the Heptagonal Number Hep(113)## 31878:

the Triangular Number T(252)## 32016:

the Nonagonal Number N(96)## 32041:

a Square, S(179)## 32131:

the Triangular Number T(253), also the Triangular Jonathan Number T(T(22))## 32319:

the Heptagonal Number Hep(114)## 32385:

the Triangular Number T(254)## 32400:

a Square, S(180)## 32640:

a Constructable Number because (2^7)x3x5x17, a power of two times a product of Fermat Primes; also the Triangular Number T(255)## 32689:

the Nonagonal Number N(97)## 32761:

a Square, S(181)## 32768:

the Cube C(32), since 32 is itself a 5th power, we have a 15th power, 2^15## 32890:

the Heptagonal Number Hep(115)## 32896:

a Constructable Number because (2^7)x257, a power of two times a product of Fermat Primes, also the Triangular Number T(256), also the Heterogenous Jonathan Number T(2^8) = T(2^(C(2)))## 33124:

a Square, S(182)## 33153:

the Triangular Number T(257)## 33320:

the Heptagonal Pyramidal Number HepPyr(34)## 33369:

the Nonagonal Number N(98)## 33411:

the Triangular Number T(258)## 33466:

the Heptagonal Number Hep(116)## 33489:

a Square, S(183)## 33551:

Truncated Tetrahedral Number TruTet(21)## 33670:

the Triangular Number T(259)## 33781:

Octahedral Number Octh(37)## 33856:

a Square, S(184)## 33930:

the Triangular Number T(260)## 34047:

the Heptagonal Number Hep(117)## 34056:

the Nonagonal Number N(99)## 34191:

the Triangular Number T(261)## 34225:

a Square, S(185)## 34447:

the Hexagonal Pyramidal Number HPyr(37)## 34453:

the Triangular Number T(262)## 34481:

Rhombic Dodecahedral Number RhoDod(21)## 34596:

a Square, S(186)## 34633:

the Heptagonal Number Hep(118)## 34716:

the Triangular Number T(263)## 34750:

the Nonagonal Number N(100), also the Nonagonal Jonathan Number N(S(10)) {to be done: append additional Nonagonal Numbers in this sequence)## 34816:

a Constructable Number because (2^11)x17 (power of two times product of Fermat Primes)## 34969:

a Square, S(187)## 34980:

the Triangular Number T(264)## 35126:

the Stella Octangula Number StOct(26)## 35224:

the Heptagonal Number Hep(119)## 35245:

the Triangular Number T(265)## 35344:

a Square, S(188)## 35511:

the Triangular Number T(266)## 35721:

a Square, S(189)## 35778:

the Triangular Number T(267)## 35820:

the Heptagonal Number Hep(120)## 35937:

the Cube C(33)## 35960:

the Pentatope Number Ptop(29)## 36046:

the Triangular Number T(268)## 36100:

a Square, S(190), also the Heterogeneous Jonathan Number S(T(19))## 36315:

the Triangular Number T(269)## 36330:

the Heptagonal Pyramidal Number HepPyr(35)## 36421:

the Heptagonal Number Hep(121), also the Heterogeneous Jonathan Number Hep(S(11))## 36481:

a Square, S(191)## 36585:

the Triangular Number T(270)## 36594:

Octahedral Number Octh(38)## 36856:

the Triangular Number T(271)## 36864:

a Square, S(192)## 37027:

the Heptagonal Number Hep(122)## 37128:

the Triangular Number T(272)## 37249:

a Square, S(193)## 37297:

the Hexagonal Pyramidal Number HPyr(38)## 37401:

the Triangular Number T(273)## 37636:

a Square, S(194)## 37638:

the Heptagonal Number Hep(123)## 37675:

the Triangular Number T(274)## 37766:

the Truncated Octahedral Number TruOct(14)## 37950:

the Triangular Number T(275)## 38025:

a Square, S(195)## 38226:

the Triangular Number T(276), also the Triangular Jonathan Number T(2,25) = T(T(25)), also the Heterogeneous Jonathan Number T(T(S(5)))## 38254:

the Heptagonal Number Hep(124)## 38416:

a Square, S(196), but since 196 is a Square, we have the Biquadratic 14^4## 38503:

the Triangular Number T(277)## 38676:

Truncated Tetrahedral Number TruTet(22)## 38781:

the Triangular Number T(278)## 38809:

a Square, S(197)## 38875:

the Heptagonal Number Hep(125), also the Heterogeneous Jonathan Number Hep(C(5))## 39060:

the Triangular Number T(279)## 39204:

a Square, S(198)## 39304:

the Cube C(34)## 39339:

the Stella Octangula Number StOct(27)## 39340:

the Triangular Number T(280)## 39501:

the Heptagonal Number Hep(126)## 39516:

the Heptagonal Pyramidal Number HepPyr(36)## 39559:

Octahedral Number Octh(39)## 39601:

a Square, S(199)## 39621:

the Triangular Number T(281)## 39775:

Rhombic Dodecahedral Number RhoDod(22)## 39903:

the Triangular Number T(282)## 40000:

a Square, S(200)## 40132:

the Heptagonal Number Hep(127)## 40186:

the Triangular Number T(283)## 40300:

the Hexagonal Pyramidal Number HPyr(39) {to be done: append additional numbers in this sequence}## 40401:

a Square, S(201)## 40470:

the Triangular Number T(284)## 40755:

the Triangular Number T(285)## 40768:

the Heptagonal Number Hep(128), also the Heterogeneous Jonathan Number Hep(7thPower(2))## 40804:

a Square, S(202)## 40920:

the Pentatope Number Ptop(30)## 40960:

a Constructable Number because (2^13)x5 (power of two times product of Fermat Primes)## 41041:

the Triangular Number T(286), also a Carmichael Number, also T(T(286)) is a Carmichael Number [as I discovered Jan 2004]## 41120:

a Constructable Number because (2^5)x5x257, a power of two times a product of Fermat Primes## 41209:

a Square, S(203)## 41328:

the Triangular Number T(287)## 41409:

the Heptagonal Number Hep(129)## 41616:

the Triangular Number T(288), also a Square, S(204), hence a Square Triangle...## 41905:

the Triangular Number T(289)## 42025:

a Square, S(205)## 42055:

the Heptagonal Number Hep(130)## 42195:

the Triangular Number T(290)## 42436:

a Square, S(206)## 42486:

the Triangular Number T(291)## 42680:

Octahedral Number Octh(40) {to be done: append additional numbers in this sequence}## 42706:

the Heptagonal Number Hep(131)## 42778:

the Triangular Number T(292)## 42849:

a Square, S(207)## 42875:

the Cube C(35)## 42883:

the Heptagonal Pyramidal Number HepPyr(37)## 43071:

the Triangular Number T(293)## 43264:

a Square, S(208)## 43362:

the Heptagonal Number Hep(132)## 43365:

the Triangular Number T(294)## 43520:

a Constructable Number because (2^9)x5x17, a power of two times a product of Fermat Primes## 43660:

the Triangular Number T(295)## 43681:

a Square, S(209)## 43690:

a Constructable Number because 2x5x17x257, a power of two times a product of Fermat Primes## 43876:

the Stella Octangula Number StOct(28)## 43956:

the Triangular Number T(296)## 44023:

the Heptagonal Number Hep(133)## 44100:

a Square, S(210), also the Heterogeneous Jonathan Number S(T(20))## 44253:

the Triangular Number T(297)## 44298:

Truncated Tetrahedral Number TruTet(23)## 44521:

a Square, S(211)## 44551:

the Triangular Number T(298)## 44689:

the Heptagonal Number Hep(134)## 44850:

the Triangular Number T(299)## 44944:

a Square, S(212)## 45150:

the Triangular Number T(300), also the Triangular Jonathan Number T(T(26))## 45360:

the Heptagonal Number Hep(135)## 45369:

a Square, S(213)## 45451:

the Triangular Number T(301)## 45585:

Rhombic Dodecahedral Number RhoDod(23)## 45753:

the Triangular Number T(302)## 45796:

a Square, S(214)## 46036:

the Heptagonal Number Hep(136), note that last two digits of index and equal to those of Heptagonal## 46056:

the Triangular Number T(303)## 46225:

a Square, S(215)## 46360:

the Triangular Number T(304)## 46376:

the Pentatope Number Ptop(31)## 46436:

the Heptagonal Pyramidal Number HepPyr(38)## 46656:

the Cube C(36), since 36 is itself a square, we have a 6th power, 6^6, and also the Square, S(216)## 46665:

the Triangular Number T(305)## 46717:

the Heptagonal Number Hep(137)## 46929:

the Truncated Octahedral Number TruOct(15)## 46971:

the Triangular Number T(306)## 47089:

a Square, S(217)## 47278:

the Triangular Number T(307)## 47403:

the Heptagonal Number Hep(138)## 47524:

a Square, S(218)## 47586:

the Triangular Number T(308)## 47895:

the Triangular Number T(309)## 47961:

a Square, S(219)## 48094:

the Heptagonal Number Hep(139)## 48205:

the Triangular Number T(310)## 48400:

a Square, S(220)## 48516:

the Triangular Number T(311)## 48749:

the Stella Octangula Number StOct(29)## 48790:

the Heptagonal Number Hep(140)## 48828:

the Triangular Number T(312)## 48841:

a Square, S(221)## 49141:

the Triangular Number T(313)## 49152:

a Constructable Number because (2^14)x3 (power of two times product of Fermat Primes)## 49284:

a Square, S(222)## 49344:

a Constructable Number because (2^6)x3x257, a power of two times a product of Fermat Primes## 49455:

the Triangular Number T(314)## 49491:

the Heptagonal Number Hep(141)## 49729:

a Square, S(223)## 49770:

the Triangular Number T(315)## 50086:

the Triangular Number T(316)## 50176:

a Square, S(224)## 50180:

the Heptagonal Pyramidal Number HepPyr(39)## 50197:

the Heptagonal Number Hep(142)## 50403:

the Triangular Number T(317)## 50440:

Truncated Tetrahedral Number TruTet(24)## 50625:

a Square, S(225), but since 225 is itself a Square, this is the Biquadratic 15^4## 50653:

the Cube C(37)## 50721:

the Triangular Number T(318)## 50908:

the Heptagonal Number Hep(143)## 51040:

the Triangular Number T(319)## 51076:

a Square, S(226)## 51360:

the Triangular Number T(320)## 51529:

a Square, S(227)## 51624:

the Heptagonal Number Hep(144), also the Heterogeneous Jonathan Number Hep(S(12))## 51681:

the Triangular Number T(321)## 51935:

Rhombic Dodecahedral Number RhoDod(24)## 51984:

a Square, S(228)## 52003:

the Triangular Number T(322)## 52224:

a Constructable Number because (2^10)x3x17, a power of two times a product of Fermat Primes## 52326:

the Triangular Number T(323)## 52345:

the Heptagonal Number Hep(145), note the last 4 digits are consecutive, and that the last two digits of the index equals the last two digits of the Heptagonal (base 10)## 52360:

the Pentatope Number Ptop(32)## 52428:

a Constructable Number because (2^2)x3x17x257, a power of two times a product of Fermat Primes## 52441:

a Square, S(229)## 52650:

the Triangular Number T(324), also the Heterogeneous Jonathan Number T(S(18))## 52900:

a Square, S(230)## 52975:

the Triangular Number T(325), also the Triangular Jonathan Number T(2,25) = T(T(25)) also the Heterogeneous Jonathan Number T(S(5))## 53071:

the Heptagonal Number Hep(146)## 53301:

the Triangular Number T(326)## 53361:

a Square, S(231), also the Heterogeneous Jonathan Number S(T(21)), also the Heterogeneous Jonathan Number S(T(T(6))), also the Heterogeneous Jonathan Number S(T(T(T(3)))), also the Heterogeneous Jonathan Number S(T(T(T(T(2))))), which we can write as S(T(4,2))## 53628:

the Triangular Number T(327)## 53802:

the Heptagonal Number Hep(147)## 53824:

a Square, S(232)## 53956:

the Triangular Number T(328)## 53970:

the Stella Octangula Number StOct(30)## 54120:

the Heptagonal Pyramidal Number HepPyr(40) {to be done: append additional numbers in this sequence}## 54285:

the Triangular Number T(329)## 54289:

a Square, S(233)## 54538:

the Heptagonal Number Hep(148), also the Heptagonal Jonathan Number Hep(Hep(8))## 54615:

the Triangular Number T(330)## 54756:

a Square, S(234)## 54872:

the Cube C(38)## 54946:

the Triangular Number T(331)## 55225:

a Square, S(235)## 55278:

the Triangular Number T(332)## 55279:

the Heptagonal Number Hep(149)## 55611:

the Triangular Number T(333)## 55696:

a Square, S(236)## 55945:

the Triangular Number T(334)## 56025:

the Heptagonal Number Hep(150)## 56169:

a Square, S(237)## 56280:

the Triangular Number T(335)## 56616:

the Triangular Number T(336)## 56644:

a Square, S(238)## 56776:

the Heptagonal Number Hep(151)## 56953:

the Triangular Number T(337)## 57121:

a Square, S(239)## 57125:

Truncated Tetrahedral Number TruTet(25)## 57291:

the Triangular Number T(338)## 57466:

the Truncated Octahedral Number TruOct(16)## 57532:

the Heptagonal Number Hep(152)## 57600:

a Square, S(240)## 57630:

the Triangular Number T(339)## 57970:

the Triangular Number T(340)## 58081:

a Square, S(241)## 58293:

the Heptagonal Number Hep(153)## 58311:

the Triangular Number T(341)## 58564:

a Square, S(242)## 58653:

the Triangular Number T(342)## 58849:

Rhombic Dodecahedral Number RhoDod(25)## 58905:

the Pentatope Number Ptop(33)## 58996:

the Triangular Number T(343), also the Heterogeneous Jonathan Number T(C(7))## 59049:

a Square, S(243)## 59059:

the Heptagonal Number Hep(154), note the first two digits equal the last two digits (base 10)## 59319:

the Cube C(39)## 59340:

the Triangular Number T(344)## 59536:

a Square, S(244)## 59551:

the Stella Octangula Number StOct(31)## 59685:

the Triangular Number T(345)## 59830:

the Heptagonal Number Hep(155)## 60025:

a Square, S(245)## 60031:

the Triangular Number T(346)## 60378:

the Triangular Number T(347)## 60516:

a Square, S(246)## 60606:

the Heptagonal Number Hep(156), also alternating same two digits (base 10), also a Palindrome (base 10)## 60726:

the Triangular Number T(348)## 61009:

a Square, S(247)## 61075:

the Triangular Number T(349)## 61387:

the Heptagonal Number Hep(157)## 61425:

the Triangular Number T(350)## 61440:

a Constructable Number because (2^12)x3x5 (power of two times product of Fermat Primes)## 61504:

a Square, S(248)## 61680:

a Constructable Number because (2^4)x3x5x257, a power of two times a product of Fermat Primes## 61776:

the Triangular Number T(351), also the Triangular Jonathan Number T(2,26) = T(T(26))## 62001:

a Square, S(249)## 62128:

the Triangular Number T(352)## 62173:

the Heptagonal Number Hep(158)## 62481:

the Triangular Number T(353)## 62964:

the Heptagonal Number Hep(159)## 62500:

a Square, S(250)## 62835:

the Triangular Number T(354)## 63001:

a Square, S(251)## 63190:

the Triangular Number T(355)## 63504:

a Square, S(252)## 63546:

the Triangular Number T(356)## 63760:

the Heptagonal Number Hep(160)## 63903:

the Triangular Number T(357)## 64000:

the Cube C(40)## 64009:

a Square, S(253), also the Heterogeneous Jonathan Number S(T(22))## 64261:

the Triangular Number T(358)## 64376:

Truncated Tetrahedral Number TruTet(26)## 64516:

a Square, S(254)## 64561:

the Heptagonal Number Hep(161)## 64620:

the Triangular Number T(359)## 64980:

the Triangular Number T(360)## 65025:

a Square, S(255)## 65280:

a Constructable Number because (2^8)x3x5x17, a power of two times a product of Fermat Primes## 65341:

the Triangular Number T(361), also the Heterogeneous Jonathan Number T(S(19))## 65367:

the Heptagonal Number Hep(162)## 65504:

the Stella Octangula Number StOct(32)## 65535:

a Constructable Number because 3x5x17x257, a power of two (here 2^0=1) times a product of Fermat Primes## 65536:

a Square, S(256), but because 256 is itself an 8th power, this is a 16th power 2^16, and hence this is also an 8th power 4^8 and hence also a Biquadratic 16^4## 65537:

a Constructable Number because a Fermat Prime## 65703:

the Triangular Number T(362)## 65792:

a Constructable Number because (2^8)x257, a power of two times a product of Fermat Primes## 66045:

the Pentatope Number Ptop(29)## 66049:

a Square, S(257)## 66066:

the Triangular Number T(363), also a Palindrome whose index as a Triangular Number is a Palindrome (base 10)## 66178:

the Heptagonal Number Hep(163)## 66351:

Rhombic Dodecahedral Number RhoDod(26)## 66430:

the Triangular Number T(364)## 66564:

a Square, S(258)## 66795:

the Triangular Number T(365)## 66994:

the Heptagonal Number Hep(164)## 67081:

a Square, S(259)## 67161:

the Triangular Number T(366)## 67528:

the Triangular Number T(367)## 67600:

a Square, S(260)## 67815:

the Heptagonal Number Hep(165), note last two digits of index are first and last digit of Heptagonal (base 10)## 67896:

the Triangular Number T(368)## 68121:

a Square, S(261)## 68265:

the Triangular Number T(369)## 68635:

the Triangular Number T(370)## 68641:

the Heptagonal Number Hep(166)## 68644:

a Square, S(262)## 68921:

the Cube C(41)## 69006:

the Triangular Number T(371)## 69169:

a Square, S(263)## 69378:

the Triangular Number T(372)## 69472:

the Heptagonal Number Hep(167)## 69473:

the Truncated Octahedral Number TruOct(17)## 69632:

a Constructable Number because (2^12)x17 (power of two times product of Fermat Primes)## 69696:

a Square, S(264), note how the digits alternate## 69751:

the Triangular Number T(373)## 69904:

a Constructable Number because (2^4)x17x257, a power of two times a product of Fermat Primes## 70125:

the Triangular Number T(374)## 70225:

a Square, S(265)## 70308:

the Heptagonal Number Hep(168)## 70500:

the Triangular Number T(375)## 70756:

a Square, S(266)## 70876:

the Triangular Number T(376)## 71149:

the Heptagonal Number Hep(169), also the Heterogeneous Jonathan Number Hep(S(13))## 71253:

the Triangular Number T(377)## 71289:

a Square, S(267)## 71631:

the Triangular Number T(378)## 71824:

a Square, S(268)## 71841:

the Stella Octangula Number StOct(33)## 71995:

the Heptagonal Number Hep(170)## 72010:

the Triangular Number T(379)## 72216:

Truncated Tetrahedral Number TruTet(27)## 72361:

a Square, S(269)## 72390:

the Triangular Number T(380)## 72771:

the Triangular Number T(381)## 72846:

the Heptagonal Number Hep(171)## 72900:

a Square, S(270)## 73153:

the Triangular Number T(382)## 73441:

a Square, S(271)## 73536:

the Triangular Number T(383)## 73702:

the Heptagonal Number Hep(172), note last two digits of index equal first and last digit of Heptagonal (base 10)## 73815:

the Pentatope Number Ptop(35) also the Pentatope Jonathan Number Ptop(Ptop(4))## 73920:

the Triangular Number T(384)## 73984:

a Square, S(272)## 74088:

the Cube C(42)## 74305:

the Triangular Number T(385)## 74465:

Rhombic Dodecahedral Number RhoDod(27)## 74529:

a Square, S(273)## 74563:

the Heptagonal Number Hep(173), note that this is the 5 consecutive digits 34567 permuted (base 10), that is to say, a Straight as a Poker hand## 74691:

the Triangular Number T(386)## 75076:

a Square, S(274)## 75078:

the Triangular Number T(387)## 75429:

the Heptagonal Number Hep(174)## 75466:

the Triangular Number T(388)## 75625:

a Square, S(275)## 75855:

the Triangular Number T(389)## 76176:

a Square, S(276), also the Heterogeneous Jonathan Number S(T(23))## 76245:

the Triangular Number T(390)## 76300:

the Heptagonal Number Hep(175)## 76636:

the Triangular Number T(391)## 76729:

a Square, S(277)## 77028:

the Triangular Number T(392)## 77176:

the Heptagonal Number Hep(176), note last 3 digits of index equals last 3 digits of Heptagonal (base 10)## 77284:

a Square, S(278)## 77421:

the Triangular Number T(393)## 77815:

the Triangular Number T(394)## 77841:

a Square, S(279)## 78057:

the Heptagonal Number Hep(177)## 78210:

the Triangular Number T(395)## 78400:

a Square, S(280)## 78574:

the Stella Octangula Number StOct(34)## 78606:

the Triangular Number T(396)## 78943:

the Heptagonal Number Hep(178), note last two digits of index equals first two digits of Heptagonal (base 10)## 78961:

a Square, S(281)## 79003:

the Triangular Number T(397)## 79401:

the Triangular Number T(398)## 79507:

the Cube C(43)## 79524:

a Square, S(282)## 79800:

the Triangular Number T(399)## 79834:

the Heptagonal Number Hep(179)## 80089:

a Square, S(283)## 80200:

the Triangular Number T(400)## 80656:

a Square, S(284)## 80601:

the Triangular Number T(401)## 80668:

Truncated Tetrahedral Number TruTet(28)## 80730:

the Heptagonal Number Hep(180), note last two digits of index equal first and last digit of Heptagonal (base 10)## 81003:

the Triangular Number T(402)## 81225:

a Square, S(285)## 81406:

the Triangular Number T(403)## 81631:

the Heptagonal Number Hep(181)## 81796:

a Square, S(286)## 81810:

the Triangular Number T(404)## 81920:

a Constructable Number because (2^14)x5 (power of two times product of Fermat Primes)## 82215:

the Triangular Number T(405)## 82240:

a Constructable Number because (2^6)x5x257, a power of two times a product of Fermat Primes## 82251:

the Pentatope Number Ptop(31)## 82369:

a Square, S(287)## 82537:

the Heptagonal Number Hep(182)## 82621:

the Triangular Number T(406)## 82944:

a Square, S(288)## 83028:

the Triangular Number T(407)## 83046:

the Truncated Octahedral Number TruOct(18)## 83215:

Rhombic Dodecahedral Number RhoDod(28)## 83436:

the Triangular Number T(408)## 83448:

the Heptagonal Number Hep(183), note last two digits of index equal first two digits of heptagonal (base 10)## 83521:

a Square, S(289), but since 289 is itself a Sqaure, this is the Biquadratic 17^4## 83845:

the Triangular Number T(409)## 84100:

a Square, S(290)## 84255:

the Triangular Number T(410)## 84364:

the Heptagonal Number Hep(184), note last two digits of index equal first two digits of Heptagonal (base 10)## 84666:

the Triangular Number T(411)## 84681:

a Square, S(291)## 85078:

the Triangular Number T(412)## 85184:

the Cube C(44)## 85285:

the Heptagonal Number Hep(185), note last two digits of index equal first two digits and last two digits of Heptagonal (base 10)## 85491:

the Triangular Number T(413)## 85624:

a Square, S(292)## 85715:

the Stella Octangula Number StOct(35)## 85849:

a Square, S(293)## 85905:

the Triangular Number T(414)## 86211:

the Heptagonal Number Hep(186), note last two digits of index equal first two digits of heptagonal (base 10)## 86320:

the Triangular Number T(415)## 86436:

a Square, S(294)## 86736:

the Triangular Number T(416)## 87025:

a Square, S(295)## 87040:

a Constructable Number because (2^10)x5x17, a power of two times a product of Fermat Primes## 87142:

the Heptagonal Number Hep(187), note last two digits of index equal first two digits of heptagonal (base 10)## 87153:

the Triangular Number T(417)## 87380:

a Constructable Number because (2^2)x5x17x257, a power of two times a product of Fermat Primes## 87571:

the Triangular Number T(418)## 87616:

a Square, S(296)## 87990:

the Triangular Number T(419)## 88078:

the Heptagonal Number Hep(188), note last two digits of index equal first two digits of heptagonal (base 10)## 88209:

a Square, S(297)## 88410:

the Triangular Number T(420)## 88804:

a Square, S(298)## 88831:

the Triangular Number T(421)## 89089:

the Heptagonal Number Hep(189), note last two digits of index equal first two digits of Heptagonal (base 10), more significantly this is also the Heptagonal Jonathan Number Hep(Hep(9))## 89253:

the Triangular Number T(422)## 89401:

a Square, S(299)## 89676:

the Triangular Number T(423)## 89755:

Truncated Tetrahedral Number TruTet(29)## 89965:

the Heptagonal Number Hep(190)## 90000:

a Square, S(300), also the Heterogeneous Jonathan Number S(T(24))## 90100:

the Triangular Number T(424)## 90525:

the Triangular Number T(425)## 90601:

a Square, S(301)## 90916:

the Heptagonal Number Hep(191)## 90951:

the Triangular Number T(426)## 91125:

the Cube C(45)## 91204:

a Square, S(302)## 91378:

the Triangular Number T(427)## 91390:

the Pentatope Number Ptop(37)## 91806:

the Triangular Number T(428)## 91809:

a Square, S(303)## 91872:

the Heptagonal Number Hep(192), note last two digits of index are first and last digits of Heptagonal (base 10)## 92235:

the Triangular Number T(429)## 92416:

a Square, S(304)## 92625:

Rhombic Dodecahedral Number RhoDod(29)## 92665:

the Triangular Number T(430)## 92833:

the Heptagonal Number Hep(193), note last two digits of index are first and last digits of Heptagonal (base 10)## 93025:

a Square, S(305)## 93096:

the Triangular Number T(431)## 93276:

the Stella Octangula Number StOct(36)## 93528:

the Triangular Number T(432)## 93636:

a Square, S(306)## 93799:

the Heptagonal Number Hep(194)## 93961:

the Triangular Number T(433)## 94249:

a Square, S(307)## 94395:

the Triangular Number T(434)## 94770:

the Heptagonal Number Hep(195)## 94830:

the Triangular Number T(435)## 94864:

a Square, S(308)## 95266:

the Triangular Number T(436)## 95481:

a Square, S(309)## 95703:

the Triangular Number T(437)## 95746:

the Heptagonal Number Hep(196), note last two digits of index equals first and last digit of Heptagonal (base 10)## 96100:

a Square, S(310)## 96141:

the Triangular Number T(438)## 96580:

the Triangular Number T(439)## 96721:

a Square, S(311)## 96727:

the Heptagonal Number Hep(197), note last two digits of index equals first and last digit of Heptagonal (base 10)## 97020:

the Triangular Number T(440)## 97336:

the Cube C(46)## 97344:

a Square, S(312)## 97461:

the Triangular Number T(441)## 97713:

the Heptagonal Number Hep(198)## 97903:

the Triangular Number T(442)## 97969:

a Square, S(313)## 98281:

the Truncated Octahedral Number TruOct(19)## 98304:

a Constructable Number because (2^15)x3 (power of two times product of Fermat Primes)## 98346:

the Triangular Number T(443)## 98596:

a Square, S(314)## 98688:

a Constructable Number because (2^7)x3x257, a power of two times a product of Fermat Primes## 98704:

the Heptagonal Number Hep(199)## 98790:

the Triangular Number T(444)## 99225:

a Square, S(315)## 99235:

the Triangular Number T(445)## 99500:

Truncated Tetrahedral Number TruTet(30)## 99681:

the Triangular Number T(446)## 99700:

the Heptagonal Number Hep(200)## 99856:

a Square, S(316)## 100128:

the Triangular Number T(447)## 100489:

a Square, S(317)## 100576:

the Triangular Number T(448)## 100701:

the Heptagonal Number Hep(201)## 101025:

the Triangular Number T(449)## 101124:

a Square, S(318)## 101269:

the Stella Octangula Number StOct(37) {to be done: append additional numbers in this sequence}## 101270:

the Pentatope Number Ptop(38)## 101475:

the Triangular Number T(450)## 101707:

the Heptagonal Number Hep(202)## 101761:

a Square, S(319)## 101926:

the Triangular Number T(451)## 102378:

the Triangular Number T(452)## 102400:

a Square, S(320) {to be done: append additional numnbers in this sequence}## 102718:

the Heptagonal Number Hep(203)## 102719:

Rhombic Dodecahedral Number RhoDod(30)## 102831:

the Triangular Number T(453)## 103285:

the Triangular Number T(454)## 103734:

the Heptagonal Number Hep(204)## 103740:

the Triangular Number T(455)## 103823:

the Cube C(47)## 104196:

the Triangular Number T(456)## 104448:

a Constructable Number because (2^11)x3x17, a power of two times a product of Fermat Primes## 104653:

the Triangular Number T(457)## 104755:

the Heptagonal Number Hep(205)## 104856:

a Constructable Number because (2^3)x3x17x257, a power of two times a product of Fermat Primes## 105111:

the Triangular Number T(458)## 105570:

the Triangular Number T(459)## 105781:

the Heptagonal Number Hep(206)## 106030:

the Triangular Number T(460)## 106491:

the Triangular Number T(461)## 106812:

the Heptagonal Number Hep(207)## 106953:

the Triangular Number T(462)## 107416:

the Triangular Number T(463)## 107848:

the Heptagonal Number Hep(208)## 107880:

the Triangular Number T(464)## 108345:

the Triangular Number T(465)## 108811:

the Triangular Number T(466)## 108889:

the Heptagonal Number Hep(209)## 109278:

the Triangular Number T(467)## 109746:

the Triangular Number T(468)## 109926:

Truncated Tetrahedral Number TruTet(31)## 109935:

the Heptagonal Number Hep(210), also the Heterogeneous Jonathan Number Hep(P(20))## 110215:

the Triangular Number T(469)## 110592:

the Cube C(48)## 110685:

the Triangular Number T(470)## 110986:

the Heptagonal Number Hep(211)## 111156:

the Triangular Number T(471)## 111628:

the Triangular Number T(472)## 111930:

the Pentatope Number Ptop(39)## 112042:

the Heptagonal Number Hep(212)## 112101:

the Triangular Number T(473)## 112575:

the Triangular Number T(474)## 113050:

the Triangular Number T(475)## 113521:

Rhombic Dodecahedral Number RhoDod(31)## 113526:

the Triangular Number T(476)## 114003:

the Triangular Number T(477)## 114481:

the Triangular Number T(478)## 114960:

the Triangular Number T(479)## 115274:

the Truncated Octahedral Number TruOct(20)## 115440:

the Triangular Number T(480)## 115921:

the Triangular Number T(481)## 116403:

the Triangular Number T(482)## 116886:

the Triangular Number T(483)## 117370:

the Triangular Number T(484)## 117649:

the Cube C(49)## 117855:

the Triangular Number T(485)## 118341:

the Triangular Number T(486)## 118828:

the Triangular Number T(487)## 119316:

the Triangular Number T(488)## 119805:

the Triangular Number T(489)## 120295:

the Triangular Number T(490)## 120786:

the Triangular Number T(491)## 121056:

Truncated Tetrahedral Number TruTet(32)## 121278:

the Triangular Number T(492)## 121771:

the Triangular Number T(493)## 122265:

the Triangular Number T(494)## 122760:

the Triangular Number T(495)## 122880:

a Constructable Number because (2^13)x3x5 (power of two times product of Fermat Primes)## 123256:

the Triangular Number T(496)## 123360:

a Constructable Number because (2^5)x3x5x257, a power of two times a product of Fermat Primes## 123410:

the Pentatope Number Ptop(40), also xxx## 123753:

the Triangular Number T(497)## 124251:

the Triangular Number T(498)## 124750:

the Triangular Number T(499)## 125000:

the Cube C(50)## 125055:

Rhombic Dodecahedral Number RhoDod(32)## 125250:

the Triangular Number T(500) {to be done: append additional Trinagular Numbers in this sequence}## 130560:

a Constructable Number because (2^9)x3x5x17, a power of two times a product of Fermat Primes## 131070:

a Constructable Number because 2x3x5x17x257, a power of two times a product of Fermat Primes## 131074:

a Constructable Number because 2x65537, a power of two times a product of Fermat Primes## 131584:

a Constructable Number because (2^9)x257, a power of two times a product of Fermat Primes## 132651:

the Cube C(51)## 132913:

Truncated Tetrahedral Number TruTet(33)## 134121:

the Truncated Octahedral Number TruOct(21)## 137345:

Rhombic Dodecahedral Number RhoDod(33)## 139264:

a Constructable Number because (2^13)x17 (power of two times product of Fermat Primes)## 139808:

a Constructable Number because (2^5)x17x257, a power of two times a product of Fermat Primes## 140608:

the Cube C(52)## 145520:

Truncated Tetrahedral Number TruTet(34)## 148877:

the Cube C(53)## 150415:

Rhombic Dodecahedral Number RhoDod(34)## 154918:

the Truncated Octahedral Number TruOct(22)## 157464:

the Cube C(54)## 158900:

Truncated Tetrahedral Number TruTet(35)## 163840:

a Constructable Number because (2^15)x5 (power of two times product of Fermat Primes)## 164289:

Rhombic Dodecahedral Number RhoDod(35)## 164480:

a Constructable Number because (2^7)x5x257, a power of two times a product of Fermat Primes## 166375:

the Cube C(55)## 174080:

a Constructable Number because (2^11)x5x17, a power of two times a product of Fermat Primes## 174760:

a Constructable Number because (2^3)x5x17x257, a power of two times a product of Fermat Primes## 175616:

the Cube C(56)## 177761:

the Truncated Octahedral Number TruOct(23)## 178991:

Rhombic Dodecahedral Number RhoDod(36)## 185193:

the Cube C(57)## 195112:

the Cube C(58)## 196608:

a Constructable Number because (2^16)x3 (power of two times product of Fermat Primes)## 196611:

a Constructable Number because 3x65537, a power of two (here 2^0=1) times a product of Fermat Primes## 197376:

a Constructable Number because (2^8)x3x257, a power of two times a product of Fermat Primes## 202746:

the Truncated Octahedral Number TruOct(24)## 205379:

the Cube C(59)## 208896:

a Constructable Number because (2^12)x3x17, a power of two times a product of Fermat Primes## 216000:

the Cube C(60)## 226981:

the Cube C(61)## 229969:

the Truncated Octahedral Number TruOct(25)## 238328:

the Cube C(62)## 246720:

a Constructable Number because (2^6)x3x5x257, a power of two times a product of Fermat Primes## 250047:

the Cube C(63)## 259526:

the Truncated Octahedral Number TruOct(26)## 261120:

a Constructable Number because (2^10)x3x5x17, a power of two times a product of Fermat Primes## 262140:

a Constructable Number because (2^2)x3x5x17x257, a power of two times a product of Fermat Primes## 262144:

the Cube C(64), since 64 is a 6th power, this is an 18th power 2^18, hence also the square of 2^9## 262148:

a Constructable Number because (2^2)x65537, a power of two times a product of Fermat Primes## 263168:

a Constructable Number because (2^10)x257, a power of two times a product of Fermat Primes## 274625:

the Cube C(65)## 278528:

a Constructable Number because (2^14)x17 (power of two times product of Fermat Primes)## 279616:

a Constructable Number because (2^6)x17x257, a power of two times a product of Fermat Primes## 287496:

the Cube C(66)## 291513:

the Truncated Octahedral Number TruOct(27)## 300763:

the Cube C(67)## 314432:

the Cube C(68)## 326026:

the Truncated Octahedral Number TruOct(28)## 327680:

a Constructable Number because (2^16)x5 (power of two times product of Fermat Primes)## 327685:

a Constructable Number because 5x65537, a power of two (here 2^0=1) times a product of Fermat Primes## 328509:

the Cube C(69)## 328960:

a Constructable Number because (2^8)x5x257, a power of two times a product of Fermat Primes## 343000:

the Cube C(70)## 349520:

a Constructable Number because (2^4)x5x17x257, a power of two times a product of Fermat Primes## 363161:

the Truncated Octahedral Number TruOct(29)## 393222:

a Constructable Number because 2x3x65537, a power of two times a product of Fermat Primes## 403014:

the Truncated Octahedral Number TruOct(30)## 445681:

the Truncated Octahedral Number TruOct(31)## 491258:

the Truncated Octahedral Number TruOct(32) {to be done: append additional numbers in this sequence; 6 more and I'll reach Truncated Octahedral Jonathan Number TruOct(TruOct(2))## 655370:

a Constructable Number because 2x5x65537, a power of two times a product of Fermat Primes## 657920:

a Constructable Number because (2^9)x5x257, a power of two times a product of Fermat Primes## 983055:

a Constructable Number because 3x5x65537, a power of two (here 2^0=1) times a product of Fermat Primes## 998991:

the Triangular Number T(1413)## 1000000:

the Square S(1000)## 1114129:

a Constructable Number because 17x65537, a power of two (here 2^0=1) times a product of Fermat Primes## 2097212:

a Constructable Number because (2^4)x3x17x257, a power of two times a product of Fermat Primes## 2457600:

a Constructable Number because (2^14)x3x5 (power of two times product of Fermat Primes)## 3342387:

a Constructable Number because 3x17x65537, a power of two (here 2^0=1) times a product of Fermat Primes## 5570645:

a Constructable Number because 5x17x65537, a power of two (here 2^0=1) times a product of Fermat Primes## 16843009:

a Constructable Number because 257x65537, a power of two (here 2^0=1) times a product of Fermat Primes Jonathan Vos Post is a Professor of Mathematics at Woodbury University in Burbank, California. His first degree in Mathematics was from Caltech in 1973. He is also, or has been also, a Professor of Astronomy at Cypress College in Orange County, California; Professor of Computer Science at California State University, Los Angeles; and Professor of English Composition at Pasadena City College. He is a widely published author of Science Fiction, Science, Poetry, Drama, and other fields. In his so-called spare time, he wins elections for local political offices and produces operas, as Secretary of Euterpe Opera Theare.Which N-gons up to N-gon(R) are in this table?The above table is infinitely incomplete. It does, however, as of 9 April 2004, include: * All Triangular Numbers through T(500) = 125250 * All Square Numbers through S(320) = 102400 * All Pentagonal Numbers through P(101) = 15251 * All Hexagonal Numbers through H(??) = ?? * All Heptagonal Numbers through Hep(212) = 112042 * All Octagonal Numbers through O(100) = 29800 * All Nonagonal Numbers through N(100) = 34750 * All Decagonal Numbers through D(50) = 9850 * All Hendecagonal Numbers through 11-gon(50) = 11075 * All Dodecagonal Numbers through 12-gon(??) = ?? * All 13-gonal Numbers through 13gon(50) = 13525 * All 14-gonal Numbers through 14gon(50) = 14750 * All 15-gonal Numbers through 15gon(50) = 15975 * All 16-gonal Numbers through 16gon(40) = 10960 * All 17-gonal Numbers through 17gon(40) = 11740 * All 18-gonal Numbers through 18gon(40) = 12520 * All 19-gonal Numbers through 19gon(40) = 13300 * All 20-gonal Numbers through 20gon(40) = 14080 * All 21-gonal Numbers through 21gon(40) = 14860 * All 22-gonal Numbers through 22gon(40) = 15640 * All 23-gonal Numbers through 23gon(40) = 16420 * All 24-gonal Numbers through 24gon(40) = 17200 * All 25-gonal Numbers through 25gon(30) = 10035 * All 26-gonal Numbers through 26gon(30) = 10470 * All 27-gonal Numbers through 27gon(30) = 10905 * All 28-gonal Numbers through 28gon(30) = 11340 * All 29-gonal Numbers through 29gon(30) = 11775 * All 30-gonal Numbers through 30gon(30) = 12210 * All 31-gonal Numbers through 31gon(31) = 13516 * All 32-gonal Numbers through 32gon(32) = 14912 * All 33-gonal Numbers through 33gon(33) = 16401 * All 34-gonal Numbers through 34gon(30) = 13950 * All 35-gonal Numbers through 35gon(30) = 14385 * All 36-gonal Numbers through 36gon(30) = 14820 * All 37-gonal Numbers through 37gon(30) = 15255 * All 38-gonal Numbers through 38gon(30) = 15690 * All 39-gonal Numbers through 39gon(30) = 16125 * All 40-gonal Numbers through 40gon(30) = 16560 * All 41-gonal Numbers through 41gon(30) = 16995 * All 42-gonal Numbers through 42gon(30) = 17430 * All 43-gonal Numbers through 43gon(30) = 17865 * All 44-gonal Numbers through 44gon(30) = 18300 * All 45-gonal Numbers through 43gon(30) = {to be done} * All 46-gonal Numbers through 44gon(30) = {to be done} * All Cube Numbers through C(70) = 343000 * All Octahedral Numbers through Octh(40) = 42680 * All Square Pyramidal Number through SPyr(?)= ??? * All Pentagonal Pyramidal Number through PPyr(??) = ?? * All Hexagonal Pyramidal Number through HPyr(??) = ?? * All Heptagonal Pyramidal Number through HepPyr(40) = 54120 * All Octagonal Pyramidal Number through OPyr(??) = ?? * All Stella Octangula Numbers through StOct(37) = 101269 Still {to be done}: * Octahedral Numbers * Dodecahedral Numbers * Icosahedral Number * Cuboctahedral Numbers of the First Kind * Cuboctahedral Numbers of the Second Kind (per my paper of Mar 2004]## Hotlinks to Math Pages of Jonathan Vos Post

## Four Nines Puzzle

"The Weekly Dispatch" of 4 February 1900, which ran a puzzle column by Dudeney, introduced a problem which is still provoking interest today. That problem was the Four Nines Puzzle, based on the even older Four Fours Puzzle, which is also discussed on this web page. Jonathan's page hotlinked here includes a complete list of equations representing, with four nines, every integer up to 314, and many beyond that. There is some deep theory towards the bottom of the page.## Four π Puzzle

Similar to the Four Nines Puzzle, and Four Fours Puzzle, mentioned above,What numbers can be made with four copies of the number "pi", or π?Jonathan's page hotlinked here includes a complete list of equations representing, with four "pi", every integer up to1,000, and many beyond that. It is a useful teaching tool for geometry or algenra students to ask: "How can we construct the smaller whole numbers, under 100 for instance, using only all four copies of the number π, parentheses, and the arithmetic operators## "

We also allow the use of exponentiation, radicals (especially the square root "+","-","x","/"?sqrt"), factorial "!", and the floor function "|_N_|" and ceiling function "|-N-|".## MATHEMATICS: Fantasy and Science Fiction about Mathematics

Warning: the above-linked page is over 300 Kilobytes long, and may load slowly. That's because it has not only the listed topic, but an original encyclopedia of many other subgenres of Science Fiction and Fantasy.Coming Soon:## Iterated Triangular Numbers

[submitted to Mathematics Magazine, January 2004]Coming Soon:## Iterated Polygonal Numbers

[submitted to Mathematics Magazine, January 2004]Coming Soon:## Triangular Carmichael Numbers: The First 22 Identified

[submitted to American Mathematics Monthly, 20 February 2004]Coming Soon:## Triangular Dodecagonal Numbers

[submitted to Mathematics Magazine, 23 February 2004]Coming Soon:## Dodecagonal Squares from Fibonacci Numbers

[submitted to Fibonacci Journal, 23 February 2004]Coming Soon:## Imaginary Mass, Momentum, and Acceleration: Physical or Nonphysical?

[Proceedings of the Fifth International Conference on Complexity Science, May 2004] Co-author #1 = Andrew Carmichael Post Co-author #2 = Professor Christine Carmichael, Woodbury UniversityComing Soon:## The Evolution of Controllability in Enzyme System Dynamics

[Proceedings of the Fifth International Conference on Complexity Science, May 2004]Coming Soon:## Adaptation and Coevolution on an Emergent Global Competitive Landscape

[Proceedings of the Fifth International Conference on Complexity Science, May 2004] Author #1 = Professor Philip V. Fellman, Southern New Hampshire University Author #2 = Professor Jonathan Vos Post, Woodbury University Author #3 = Roxana Wright, Southern New Hampshire UniversityComing Soon:## My Erdos Number, and how I connect to Erdos through Two Nobel laureates in Physics.

## Teaching Resume of Jonathan Vos Post

## My Teachers' Teachers' Teachers

Some of the most famnous mathematicians in history were Jonathan's teachers of teachers of teachers... See this intellectual geneology.Coming Soon: Annotated Partial List of 65 Mathematics and Computer Science Publications and Presentations in my credits.## Hotlinks to Other Cool Math Pages

## Periodic Table of the Mathematicians

## What's Special About This Number

Delightful coloful table of almost all intgers up to 10,000 with interesting facts about each one. I have contributed about 30 of these, which Professor Erich Freidman quite reasonably does not attribute (as he is interested in facts only on this page). But for the record, the 110 ones that I invented (or rediscovered in this context) and submitted to Dr. Friedman include:## Eric Weisstein's World of Mathematics

The home page pointing to (and searchable on) thousands of pages in the single most important Mathematics Site on the World Wide Web. Pi at Mathworld Mathematics in Film Mathematics in Literature## Hotlinks to Some Other Pages of Jonathan Vos Post

There are roughly 800 web pages in the doman created and run by Jonathan and his Physics Professor wife. The home page is:## Magic Dragon Multimedia

Some pages within that domain, which gets over 1,000,000 hits per month, include:## Software/Management Resume of Jonathan Vos Post

biographical and bibliographical info on Jonathan Vos PostPERIODIC TABLE OF MYSTERY AUTHORS## AUTHORS of Ultimate Science Fiction Web Guide

9,000+ more authors indexed## AUTHORS of Ultimate Westerns Web Guide

1,000+ more authors indexed## AUTHORS of Ultimate Romance Web Guide

8,000+ more authors indexed## A Discussion of Some Deeper Mathematical Issues Related to the Four Nines Puzzle

There is, of course, no upper limit to the numbers which we can build with nines and the operators mentioned at the top of this web page. Consider the infinite series: 9 9! (9!)! ((9!)!)! (((9!)!)!)! ((((9!)!)!)!)! ... and that just uses one 9. Playing around with the puzzle, it soon becomes obvious that there are integers that can be represented by four nines in an infinite number of different ways. It is NOT obvious whether ALL numbers can be represented in at least one way. We return to this later. By more advanced mathematics, it might be shown that every integer can be represented by a sufficiently long sequence of the operators we use here.In "The Weekly Dispatch" of 4 February 1900, the puzzle column by Dudeney introduced this problem. But Professor Donald Knuth comments on Dudeney's Solution Number 310, which gives a table. Knuth criticizes: "he disallows (sqrt 9)! for completely illogical reasons; also, he fails to express 38, 41, 43, ... with fewer than five 9s." Knuth on Dudeney## Dudeney Invented the Four Nines; Knuth Criticizes Dudeney

## Let us note that there are some unsolved mathematical questions about the Factorial Function N! The well-known definition is: 1! = 1 2! = 1 x 2 = 2 3! = 1 x 2 x 3 = 6 4! = 1 x 2 x 3 x 4 = 24 5! = 1 x 2 x 3 x 4 x 5 = 120 6! = 1 x 2 x 3 x 4 x 5 x 6 = 720 7! = 1 x 2 x 3 x 4 x 5 x 6 x 7 = 5040 8! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 = 40,320 9! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 = 362,880 10! =1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 = 3,628,800 ... and so on

Definition of Factorial## Believe it or not, your telephone number, your Social Security Number, or any other whole number important to you can be found as the beginning digits of some enormous factorial. John E. Maxfield proved this theorem: "If A is any positive integer having M digits, there exists a positive integer N such that the first M digits of N! constitute the integer A." ["A Note on N!", John E. Maxfield. Kansas State University, Mathematics Magazine, March-April 1970, pp.64-67; supported in part by a National Science Foundation grant].

Any Integer is the First Digits of Some Factorial## For non-integers, and for complex numbers, the universally accepted generalization of the Factorial Function is the Gamma Function. Discussion of that shall be reserved until I provide my analysis of the Four Nines Puzzle as applied to Complex Numbers, such as first appear with sqrt (-9) = 3

Factorials of Negative, Fractional, and Complex Numbersi.## In our solutions to the Four Nines Puzzle, we often add to or subtract something from a factorial, and then take a square root. Typically, this is not a whole number, so we round down, or round up (floor or ceiling). This leads to the question: When is N! + 1 a perfect square? Or, equivalently, When is sqrt (N! + 1) a whole number? The only solutions known are: 25 = 4! + 1 = 5 x 5 121 = 5! + 1 = 11 x 11 5041 = 7! + 1 = 71 x 71 Nobody knows if there are any other perfect squares one more than factorials. Clearly related is a paper by Berend and Osgood: [Journal of Number Theory, vol. 42, 1992] which proves that for any polynomial P of degree > 1 the set of positive integers N for which P(X) = N! has an integral solution X, is of zero density. This paper explicitly says that it is not known if the equation X^2 - 1 = N! has only finitely many solutions. We also don't know if there are an infinite number of primes of the form n! + 1. We also don't know if there are an infinite number of primes of the form n! - 1. The largest such prime that we know is 3610! - 1. It has 11,277 digits [Caldwell, title to be added here, 1993] By the way, there is a fairly elementary proof that, except for 0!=1 and 1!=1, NO factorials are perfect squares. That fairly elementary proof, though, uses a heavy mathematical result known as Bertrand's Postulate, also known as Chebychev's Theorem (after the man who first proved it). This Theorem says that there always exists at least one prime between N and 2N, if N>2. Erdos gave a genuinely elementary (although neither short nor obvious) proof. This will all be inserted here or linked to in a later version of this web page.

Square-less-one Factorials## The Four Nines Puzzle itself is, to be sure, very elementary stuff. So I stand meekly in the shadow of the great mathematician's who were my Teachers' teachers' teachers... Gottfried Leibnitz, Jacob Bernoulli, Johann Bernoulli, Leonhard Euler, Joseph Louis Lagrange, Simeon Poisson, Pafnuty Lvovich Chebyshev, Andrei A. Markov, G. H. Hardy, Alonzo Church, David Hilbert, Norbert Wiener, Alan Turing... With them in mind, I remark that Donald Knuth conjectures that

Ergodic HypothesisALLintegers can be made with a sufficiently lengthy combination of square roots and factorials and floors and ceilings ... built around a single 4. I make the related conjecture, based on the number 9. As we see near the top of this web page, we can make a 4 from a single 9, with a lot of square roots, factorials, and floor functions. Hence Knuth's conjecture for 4 immediately applies to 9. In summary of a subtle proof of Knuth's Conjecture, still in progress, factorials make a number bigger, and square roots make it smaller. Iterating sufficiently, we are "folding" the algebraic number line back onto itself recursively, and this is an ergodic property, which carries a number arbitrarily close to any given integer, at which point a final floor or ceiling gets us exactly to that given integer.## The problem becomes (as I shall show in a forthcoming paper co-authored by Andrew Carmichael Post and Dr. George Hockney) how to achieve as many integers as possible with 4 nines -- for a given degree of "complexity" as defined by the number of symbols of a standardized way of expressing the combination of 9, 99, 999, 9999, +, -, x, /, sqrt, factorial, floor, and ceiling (say in Backus-Naur Form). For example: 9999 = 9999 (complexity = 4) 1 = 99/99 (complexity = 5) 1008 = 999 + 9 (complexity = 5) 9801 = 99 x 99 (complexity = 5) 2 = 99/9 - 9 (complexity = 6) 20 = 99/9 + 9 (complexity = 6) 19 = 9 + 9 + 9/9 (complexity 7) 36 = 9 + 9 + 9 + 9 (complexity 7) 13 = 9 + sqrt 9 + 9/9 (complexity 8) 40 = |_ sqrt 999 _| + 9 (complexity 8) and so on. The complexity function creates an order on the solutions to the four nines problem. Of interest are such functions as the smallest number whose complexity exceeds a given value, and upper and lower bounds on the ratio of a number to its complexity. Almost all numbers have very high complexity. But details will be revealed in that forthcoming paper. The problem of whether two strings of characters evaluate to the same integer is a very hard problem, in terms of the amount of computation necessary to determine it in general, called the "word problem" in complexity theory.

Complexity Ordering of SolutionsThe "Four Fours Problem" first appeared in: "Mathematical Recreations and Essays", by W. W. Rouse Ball [1892]. In this book the "Four Fours Problem" is called a "traditional recreation." There are several fine sites on the World Wide Web for "Four Fours Problem." But I recommend to the reader: "Mathematical Games", by Martin Gardner, [Scientific America, Jan 1964]. The [Feb 1964] issue has answers to the puzzles posed in January. Martin Gardner was extending the "two fours" problem as first posed by J. A. Tierney in 1944, and extended by others in 1964. ["E631", J. A. Tierney, Amer. Math Monthly, 52(1945)219]. ["64 Ways to Write 64 Using four 4's", M. Bicknell and V. E. Hoggatt, Recreational Mathematics Magazine, 14(1964)13-25]. More recently, we have Knuth's Conjecture: "Representing Numbers Using Only One 4", Donald Knuth, [Mathematics Magazine, Vol. 37, Nov/Dec 1964, pp.308-310]. Knuth shows how (using a computer program he wrote) all integers from 1 through 207 may be represented with only one 4, varying numbers of square roots, varying numbers of factorials, and the floor function. For example: Knuth shows how to make the number 64 using only one 4: |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt |_ sqrt |_ sqrt sqrt sqrt sqrt sqrt (4!)! _| ! _| ! _| ! _| ! _| ! _| ! _| ! _| As to notation in the above example, he means sqrt n! stands for sqrt (n!), not (sqrt n)! Knuth further points out that |_ sqrt |_ X _| _| = |_ sqrt X _| so that the floor function's brackets are only needed around the entire result and before factorials are taken. He CONJECTURES that all integers may be represented that way: "It seems plausible that all positive integers possess such a representation, but this fact (if true) seems to be tied up with very deep propertis of the integers." Your Humble Webmaster believes that Knuth is right, for 9 as well as 4, and will prove that in a forthcoming paper. Knuth comments: "The referee has suggested a stronger conjecture, that a representation may be found in which all factorial operations precede all square root operations; and, moreover, if the greatest integer function [our floor function] is not used, an arbitrary positive real number can probably be approximated as closely as desired in this manner." If we abbreviate the long strings of symbols, we can express this more elegantly. Let NK be the number (...((4!)!)!...)! with K factorial operations. Then the anonymous referee's stronger conjecdture is equivalent to: log NK/(2^|_log(log NK)_|, for K=1,2,3..., are dense in the interval (1,2). Notationally here log means log to the base 2, and ^ means exponentiation. One key to the proof I shall publish is that, for any logarithm base: "The fractional part of log N is dense on the unit interval." From that, it has been proved that: "The fractional part of log N! is dense on [0,1]." ["A Note on N!", John E. Maxfield. Kansas State University, Mathematics Magazine, March-April 1970, pp.64-67; supported in part by a National Science Foundation grant]. Maxfield also proves, in the notation above, if we also define logK is the Kth iterant of Log N, i.e. log2(N) = log log N, that: The fractional part of logK (NK) is dense on the unit interval.## The "Four Nines Problem" is closely related to the "Four Fours Problem"

## In our solutions of the Four Nines Puzzle, we sometimes take a square root of the sum of two slightly different square roots. As noted in Mathpages Mathpages #305: How can we find integer solutions of M = sqrt ( sqrt (N) + sqrt ((KxN)+1)) "This can be viewed as a Pell Equation with an extra solution on the solution." We have: M^2 = sqrt (N) + sqrt ((KxN)+1) in integers, so we know that there exist integers X and Y such that: N = Y^2 KN + 1 = X^2 X + Y = M^2 Eliminating N from the first two equations above gives the Pell equation: X^2 + KY^2 = 1 For any given K we are looking for solutions X,Y such that X+Y is a square. "Of course, for any positive integer K there are infinitely many solutions to the Pell Equation, but solutions with X+Y=square are rare. For example, with K=8, the values of X+Y satisfying the Pell Equation" are: ((16 + 11 sqrt 2)/8) (3 + sqrt 8)^Q + ((16 - 11 sqrt 2)/8) (3 - sqrt 8)^Q This gives the sequence 4, 23, 134, 781, 4552 ..., which satisfies the second-order recurrence: S(j) = 6S(j-1) - S(j-2) "So the question is whether this sequence contains any squares after the initial vale 4. Recall the proof that the only square Fibonacci numbers are 0, 1, and 144." [J. Cohn, "On the Square Fibonacci Numbers", J. London Math Soc., 39 (1964) 537-540] "In general the problem reduces to finding square terms of a general second-order recurring sequence, like the Fibonacci sequence. The best approach might be to apply Cohn's method of proof to the general second-order recurrence."

Square Root of a Sum of Square Roots## In our solutions to the "Four Nines Problem" we often take square roots of factorials, and sometimes square roots of square roots of factorial of factorials, and so forth. As it turns out, there are some solved and some unsolved problems in the Physics of Quantum Wave Functions which involve square roots of factorials. As B. Nagel has written, "Mathematical problems in Quantum Optics", {ref to be done}: "As a continuation of earlier studies of squeezed states and other special harmonic oscillator states of interest in quantum optics I have studied the phase representations of these states. Although the phase observable, which is roughly speaking conjugate to the number operator, does not exist as a hermitian operator -- this is a longstanding and still popular problem, initiated by Dirac in 1927 -- it exists as a so-called general observable and permits a probability interpretation via a phase distribution on the unit circle. The corresponding wave function is obtained simply by substituting the harmonic wave exp(in[[phi]]) for the number state |n>.

Square Roots of Factorials in Quantum MechanicsThe series expansions thus obtained for the coherent and squeezed states contain a square root of a factorial n!, which makes it impossible to get closed analytical expressions. Approximate analytical expressions have been derived, valid for large values of the mean value of the number operator...." For another example, see the computer program given in: GrozinHydrogen Wave Functions and E1 TransitionsProcedure R(n,1); % radial wave function -2/n^2*sqrt(factorial(n+1)/factorial(n-1-1))*exp(-r/n) ... etc. Wouldn't it be interesting if an in-depth analysis of the century-old recreation math puzzle about Four Nines turned out to be useful in solving a problem in 21st Century Quantum Optics with Squeezed States? {more discussion to be added in February 2004} Special Thanks to Dr. George Hockney, NASA/JPL, for informal discussion and review in January-February 2004. Thanks to Forrest Bishop for informal discussion and review in January-February 2004.

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