cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A118278 Conjectured largest number that is not the sum of three n-gonal numbers, or -1 if there is no largest number.

Original entry on oeis.org

0, -1, 33066, 146858, 273118, -1, 1274522, 2117145, 3613278, -1, 7250758, -1, 12911636, -1, 22655394, 26801303, 25049533, -1, 56922533, 115715602, 81539010, -1, 85105105, -1, 106555658, -1, 233296317, 267370631, 286763923, -1, 358322750
Offset: 3

Views

Author

T. D. Noe, Apr 21 2006

Keywords

Comments

Extensive calculations show that if a(n) >= 0, then every number greater than a(n) can be represented as the sum of three n-gonal numbers. a(3)=0 because every number can be written as the sum of three triangular numbers. When n is a multiple of 4, there is an infinite set of numbers not representable. For n=14, there appears to be a sparse, but infinite, set of numbers not representable.

Links

Crossrefs

Cf. A118279 (number of numbers not representable).
Cf. A003679 (not the sum of three pentagonal numbers).
Cf. A007536 (not the sum of three hexagonal numbers).
Cf. A213523 (not the sum of three heptagonal numbers).
Cf. A213524 (not the sum of three octagonal numbers).
Cf. A213525 (not the sum of three 9-gonal numbers).
Cf. A214419 (not the sum of three 10-gonal numbers).
Cf. A214420 (not the sum of three 11-gonal numbers).
Cf. A214421 (not the sum of three 12-gonal numbers).

Extensions

a(22)-a(33) from Donovan Johnson, Apr 17 2010

A280719 Expansion of (Sum_{k>=0} x^(k*(2*k-1)))^6.

Original entry on oeis.org

1, 6, 15, 20, 15, 6, 7, 30, 60, 60, 30, 6, 15, 60, 90, 66, 45, 60, 80, 90, 66, 50, 120, 180, 135, 60, 15, 60, 186, 210, 141, 126, 120, 126, 165, 180, 241, 300, 210, 90, 90, 180, 270, 270, 210, 212, 270, 270, 200, 210, 366, 450, 390, 270, 135, 210, 375, 360, 396, 420, 300, 330, 375, 380, 510, 480, 336, 450, 510, 390, 330
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 10 2017

Keywords

Comments

Number of ways to write n as an ordered sum of 6 hexagonal numbers (A000384).
a(n) > 0 for all n >= 0.
Every number is the sum of at most 6 hexagonal numbers.
Every number is the sum of at most k k-gonal numbers (Fermat's polygonal number theorem).

Examples

			a(6) = 7 because we have:
[6, 0, 0, 0, 0, 0]
[0, 6, 0, 0, 0, 0]
[0, 0, 6, 0, 0, 0]
[0, 0, 0, 6, 0, 0]
[0, 0, 0, 0, 6, 0]
[0, 0, 0, 0, 0, 6]
[1, 1, 1, 1, 1, 1]

Links

Crossrefs

Cf. A000384, A007536, A008440, A045848, A280718, A282248.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Sum[x^(k (2 k - 1)), {k, 0, nmax}]^6, {x, 0, nmax}], x]

Formula

G.f.: (Sum_{k>=0} x^(k*(2*k-1)))^6.

A180917 Numbers that are not the sum of three positive heptagonal numbers.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 34, 35, 38, 39, 40, 41, 44, 45, 46, 47, 49, 50, 51, 52, 55, 56, 58, 60, 61, 62, 64, 65, 66, 67, 68, 71, 72, 73, 76, 77, 78, 79, 81, 82, 84, 85, 87, 88, 92, 93, 94, 97, 98, 99, 101
Offset: 1

Views

Author

Jonathan Vos Post, Sep 23 2010

Keywords

Comments

Complement of A117105. This is to heptagonal numbers A000566,
as A007536 is to hexagonal numbers A000384,
as A003679 is to pentagonal numbers A000326,
and as A004214 is to squares A000290.
This sequence is presumably finite: what is its likely last element?
Last element appears to be a(1671) = 273118. - Charles R Greathouse IV, Sep 27 2010

Links

Crossrefs

Cf. A000290, A000326, A000384, A000566, A003679, A004214, A007536, A117105.

A117089 Primes that are not the sum of 3 hexagonal numbers.

Original entry on oeis.org

5, 11, 19, 23, 37, 41, 53, 59, 83, 89, 113, 131, 167, 173, 179, 229, 251, 269, 293, 313, 317, 383, 389, 439, 443, 509, 599, 641, 683, 859, 929, 1031, 1033, 1049, 1163, 1193, 1283, 1301, 1303, 1307, 1439, 1493, 1499, 1543, 1619, 1733, 2143, 2153, 2333, 2687, 2693, 3083, 3089, 3533, 3719, 3989, 4003, 4583, 4673, 4703, 5387, 5651, 5849, 5903, 6173, 6389, 6449, 7481, 9293, 12113, 15803, 16433, 19763, 61403
Offset: 1

Views

Author

Jonathan Vos Post, Apr 18 2006

Keywords

Examples

			5 is the sum of five hexagonal numbers; 11 is the sum of six hexagonal numbers; the other 72 primes are the sum of four hexagonal numbers. - _T. D. Noe_, Apr 20 2006

References

  • Legendre, Théorie des Nombres, 3rd edition, 1830.

Links

Crossrefs

Cf. A000040, A000384, A007527, A007536, A117065.

Programs

  • Mathematica
    nn=201; hex=Table[n(2n-1), {n,0,nn-1}]; ps=Prime[Range[PrimePi[hex[[ -1]]]]]; Do[n=hex[[i]]+hex[[j]]+hex[[k]]; If[n<=hex[[ -1]]&&PrimeQ[n], ps=DeleteCases[ps,n]], {i,nn}, {j,i,nn}, {k,j,nn}]; ps (* T. D. Noe, Apr 20 2006 *)

Formula

A000040 INTERSECT A007536.

Extensions

More terms from T. D. Noe, who conjectures that the list shown here is complete. His search up to 7*10^7 gave no further terms. - Apr 20 2006
Showing 1-4 of 4 results.

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