A003679 -id:A003679 - cp's OEIS frontend

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

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

T. D. Noe, Apr 21 2006

sign

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.

R. K. Guy, Every number is expressible as the sum of how many polygonal numbers?, Amer. Math. Monthly 101 (1994), 169-172.
Gordon Pall, Large positive integers are sums of four or five values of a quadratic function, Am. J. Math 54 (1931) 66-78
Eric Weisstein's World of Mathematics, MathWorld: Polygonal Number

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).

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

A100878 Smallest number of pentagonal numbers which sum to n.

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 1, 2, 3, 3, 4, 2, 3, 4, 4, 5, 1, 2, 2, 3, 4, 2, 3, 3, 4, 5, 3, 4, 2, 1, 2, 3, 4, 3, 2, 3, 4, 5, 2, 3, 3, 2, 3, 3, 4, 1, 2, 3, 4, 5, 2, 2, 3, 3, 4, 3, 3, 2, 3, 4, 3, 4, 3, 3, 1, 2, 3, 2, 3, 2, 3, 4, 3, 3, 3, 4, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 1, 2, 3, 3, 4, 2, 3, 4, 4, 4, 2, 3, 2

Offset: 0

Franz Vrabec, Jan 09 2005

nonn

From Bernard Schott, Jul 15 2022: (Start)
In September 1636, Fermat, in a letter to Mersenne, made the statement that every number is a sum of at most three triangular numbers, four squares, five pentagonal numbers, and so on.
The square case was proved by Lagrange in 1770; it is known as Lagrange's four squares theorem (see A002828). Then Gauss proved the triangular case in 1796 (see A061336).
In 1813, Cauchy proved this polygonal number theorem: for m >= 3, every positive integer N can be represented as a sum of m+2 (m+2)-gonal numbers, at most four of which are different from 0 and 1 (Deza reference). Hence every number is expressible as the sum of at most five positive pentagonal numbers (A000326). (End)

			a(5)=1 since 5=5, a(6)=2 since 6=1+5, a(7)=3 since 7=1+1+5, a(10)=2 since 10=5+5 with 1 and 5 pentagonal numbers.

Elena Deza and Michel Marie Deza, Fermat's polygonal number theorem, Figurate numbers, World Scientific Publishing (2012), Chapter 5, pp. 313-377.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section D3, Figurate numbers, pp. 222-228.

Augustin-Louis Cauchy, Démonstration du théorème général de Fermat sur les nombres polygones, Extrait des Mémoires de l'Institut, 1813-15.
Eric Weisstein's World of Mathematics, Fermat's Polygonal Number Theorem.
Wikipedia, Fermat polygonal number theorem.

Cf. A002828, A061336, A355717.

Cf. A000326 (a(n) = 1), A003679 (a(n) = 4 or 5), A355660 (a(n) = 4), A133929 (a(n) = 5).

a(n) = my(nb=oo); forpart(vp=n, if (vecsum(apply(x->ispolygonal(x, 5), Vec(vp))) == #vp, nb = min(nb, #vp)),,5); nb; \\ Michel Marcus, Jul 15 2022

a(n) = for(i = 1, oo, p = partitions(n, , [i,i]); for(j = 1, #p, if(sum(k = 1, i, ispolygonal(p[j][k],5)) == i, return(i)))) \\ David A. Corneth, Jul 15 2022

a(n) <= 5 (inequality proposed by Fermat and proved by Cauchy). - Bernard Schott, Jul 13 2022

More terms from David Wasserman, Mar 04 2008

A117065 Primes that are not the sum of 3 pentagonal numbers.

Original entry on oeis.org

19, 31, 43, 67, 89, 101, 113, 131, 229, 241, 277, 359, 383, 491, 523, 619, 631, 643, 701, 761, 1321, 1381, 1621, 2221, 2861

Offset: 1

Jonathan Vos Post, Apr 17 2006

5 is the only prime pentagonal number; every greater pentagonal number A000326(n) = n(3n-1)/2 is either divisible by n/2 or (3n-1)/2. Every number is the sum of 5 pentagonal numbers, hence every prime is the sum of 5 pentagonal numbers. There are an infinite number of primes which are the sum of two pentagonal numbers, the subset of primes which are the sum of two pentagonal numbers in exactly two different ways begins {211, 853, 1259, 1427, 1571, 2297, 2351}.
The sum may include the pentagonal number 0. Hence this sequence does not have any primes that are the sum of two positive pentagonal numbers. The sequence is probably finite. There are no other primes < 59900. - T. D. Noe, Apr 19 2006
The next term, if it exists, is greater than 160000000. - Jack W Grahl, Jul 10 2018
a(26) > 10^11, if it exists. - Giovanni Resta, Jul 13 2018

J. W. Grahl, C code which was used to check for elements of this sequence up to 160,000,000.
R. K. Guy, Every number is expressible as the sum of how many polygonal numbers?, Amer. Math. Monthly 101 (1994), 169-172.

Cf. A000040, A000326, A003679, A064826.

nn=201; pen=Table[n(3n-1)/2, {n,0,nn-1}]; ps=Prime[Range[PrimePi[pen[[ -1]]]]]; Do[n=pen[[i]]+pen[[j]]+pen[[k]]; If[n<=pen[[ -1]]&&PrimeQ[n], ps=DeleteCases[ps, ?(#==n&)]], {i,nn}, {j,i,nn}, {k,j,nn}]; ps (* _T. D. Noe, Apr 19 2006 *)

A000040 INTERSECT A003679.

More terms from T. D. Noe, Apr 19 2006

Mathematica program corrected by Robert Price, Aug 25 2019

A280718 Expansion of (Sum_{k>=0} x^(k(3k-1)/2))^5.

Original entry on oeis.org

1, 5, 10, 10, 5, 6, 20, 30, 20, 5, 10, 30, 35, 30, 30, 30, 25, 30, 60, 60, 25, 5, 35, 80, 70, 51, 35, 50, 80, 90, 80, 30, 35, 60, 80, 95, 90, 90, 50, 75, 140, 140, 85, 20, 70, 120, 130, 120, 95, 115, 100, 115, 140, 155, 110, 40, 80, 200, 230, 140, 81, 120, 200, 190, 180, 120, 80, 100, 160, 240, 200, 155, 120, 140, 245, 260, 230

Offset: 0

Ilya Gutkovskiy, Feb 10 2017

nonn

Number of ways to write n as an ordered sum of 5 pentagonal numbers (A000326).
a(n) > 0 for all n >= 0.
Every number is the sum of at most 5 pentagonal numbers.
Every number is the sum of at most k k-gonal numbers (Fermat's polygonal number theorem).

			a(5) = 6 because we have:
[5, 0, 0, 0, 0]
[0, 5, 0, 0, 0]
[0, 0, 5, 0, 0]
[0, 0, 0, 5, 0]
[0, 0, 0, 0, 5]
[1, 1, 1, 1, 1]

Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Pentagonal Number
Index to sequences related to polygonal numbers

Cf. A000326, A003679, A008439, A038671, A280719, A282248.

nmax = 76; CoefficientList[Series[Sum[x^(k (3 k - 1)/2), {k, 0, nmax}]^5, {x, 0, nmax}], x]

G.f.: (Sum_{k>=0} x^(k*(3*k-1)/2))^5.

A117104 Sum of two positive heptagonal numbers A000566.

Original entry on oeis.org

2, 8, 14, 19, 25, 35, 36, 41, 52, 56, 62, 68, 73, 82, 88, 89, 99, 110, 113, 115, 119, 130, 136, 146, 149, 155, 162, 166, 167, 182, 190, 193, 196, 203, 207, 223, 224, 229, 236, 242, 244, 253, 260, 269, 270, 287, 290, 293, 296, 301, 304, 316, 320, 337, 341, 343, 347

Offset: 1

Jonathan Vos Post, Apr 18 2006

7 is the only prime heptagonal number. Primes which are sums of two positive heptagonal numbers include: {2, 19, 41, 73, 89, 113, 149, 167, 193, 223, 229, 269, 293, 337, 347, 367, 383, ...}.

Cf. A000566, A000040, A000326, A003679, A064826, A117065.

Lim=348;nl=Ceiling[Sqrt[(2/5)Lim]];Select[Union[Total/@Tuples[Table[n*(5n-3)/2,{n,nl}],{2}]],#James C. McMahon, Sep 27 2024 *)

{a(n)} = {A000566} + {A000566} = {a*(5*a-3)/2 + b*(5*b-3)/2} \ {A000566}.

Missing 343 added by Giovanni Resta, Jun 15 2016

A117105 Numbers that are the sum of three positive heptagonal numbers (A000566) in at least one way.

Original entry on oeis.org

3, 9, 15, 20, 21, 26, 32, 36, 37, 42, 43, 48, 53, 54, 57, 59, 63, 69, 70, 74, 75, 80, 83, 86, 89, 90, 91, 95, 96, 100, 102, 106, 107, 111, 114, 116, 117, 120, 122, 123, 126, 128, 131, 133, 137, 143, 144, 147, 148, 149, 150, 153, 154, 156, 162, 163, 164, 165

Offset: 1

Jonathan Vos Post, Apr 18 2006

7 is the only prime heptagonal number. Primes which are sums of two positive heptagonal numbers include: {2, 19, 41, 73, 89, 113, 149, 167, 193, 223, 229, 269, 293, 337, 347, 367, 383, ...}. Primes which are sums of three positive heptagonal numbers include: {3, 37, 43, 53, 59, 83, 89, 107, 131, 137, 149, 163, 167, 173, 191, 197, 211, 227, 241, 251, 257, 263, 271, ...}.

By definition this does not contain any repeated terms. - N. J. A. Sloane, Aug 15 2020

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000566, A000040, A000326, A003679, A064826, A117065.

With[{nn=10},Select[Union[Total/@Tuples[PolygonalNumber[7,Range[ nn]],3]], #<=PolygonalNumber[7,nn]-2&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 16 2020 *)

{a(n)} = {A000566} + {A000566} + {A000566} = {a*(5*a-3)/2 + b*(5*b-3)/2 + c*(5*c-3)/2} \ {A000566}.

Missing 106 and 131 added by Giovanni Resta, Jun 15 2016

Corrected (deleting duplicates) and extended by Harvey P. Dale, Aug 16 2020

A133929 Positive integers that cannot be expressed using four pentagonal numbers.

Original entry on oeis.org

9, 21, 31, 43, 55, 89

Offset: 1

Eric W. Weisstein, Sep 29 2007

Equivalently, integers m such that the smallest number of pentagonal numbers (A000326) which sum to m is exactly five, that is, A100878(a(n)) = 5. Richard Blecksmith & John Selfridge found these six integers among the first million, they believe that they have found them all (Richard K. Guy reference). - Bernard Schott, Jul 22 2022

			   9 =  5 +  1 + 1 + 1 + 1.
  21 =  5 +  5 + 5 + 5 + 1.
  31 = 12 + 12 + 5 + 1 + 1.
  43 = 35 +  5 + 1 + 1 + 1.
  55 = 51 +  1 + 1 + 1 + 1.
  89 = 70 + 12 + 5 + 1 + 1.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section D3, Figurate numbers, pp. 222-228.

Eric Weisstein's World of Mathematics, Pentagonal Number

Cf. A000326, A007527, A100878.

Equals A003679 \ A355660.

A355660 Numbers m such that the smallest number of pentagonal numbers (A000326) which sum to m is exactly 4.

Original entry on oeis.org

4, 8, 16, 19, 20, 26, 30, 33, 38, 42, 50, 54, 60, 65, 67, 77, 81, 84, 88, 90, 96, 99, 100, 101, 111, 112, 113, 120, 125, 131, 135, 138, 142, 154, 159, 160, 166, 170, 171, 183, 195, 204, 205, 207, 217, 224, 225, 226, 229, 230, 236, 240, 241, 243, 255, 265, 275, 277, 286, 306, 308, 345

Offset: 1

Bernard Schott, Jul 12 2022

nonn

Richard Blecksmith & John Selfridge found 204 such integers among the first million, the largest of which is 33066. They believe that they have found them all (Richard K. Guy reference).

a(205) > 10^11, if it exists, from Giovanni Resta in A003679.

			4 = 1 + 1 + 1 + 1.
8 = 5 + 1 + 1 + 1.
16 = 5 + 5 + 5 + 1.
Also, it is not possible to get these terms when summing three or fewer pentagonal numbers.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section D3, Figurate numbers, pp. 222-228.

Richard K. Guy, Every number is expressible as the sum of how many polygonal numbers?, Amer. Math. Monthly 101 (1994), 169-172.

Cf. A000326, A100878.

Equals A003679 \ A133929.

nn = 100;
pen = Table[n (3n - 1)/2, {n, 0, nn - 1}];
lst = Range[pen[[-1]]];
Do[n = pen[[i]]+pen[[j]]+pen[[k]]; If[n <= pen[[-1]], lst = DeleteCases[lst, n]], {i, 1, nn}, {j, i, nn}, {k, j, nn}];
A003679 = lst;
Complement[A003679, {9, 21, 31, 43, 55, 89}] (* Jean-François Alcover, Jul 13 2022, after T. D. Noe in A003679 *)

A100878(a(n)) = 4.

A117111 Sum of four positive heptagonal numbers A000566.

Original entry on oeis.org

4, 10, 16, 21, 22, 27, 28, 33, 37, 38, 39, 43, 44, 49, 50, 54, 55, 58, 60, 61, 64, 66, 70, 71, 72, 75, 76, 77, 81, 82, 84, 87, 88, 90, 91, 92, 93, 96, 97, 98, 101, 102, 103, 104, 107, 108, 109, 112, 113, 114, 115, 117, 118, 120, 121, 123, 124, 125, 127, 129, 130, 132

Offset: 1

Jonathan Vos Post, Apr 18 2006

Fermat discovered, Gauss, Legendre and [1813] Cauchy proved that every integer is the sum of 7 heptagonal numbers (and there are some numbers which require all 7, the smallest being 13). 7 is the only prime heptagonal number. Primes which are sums of two positive heptagonal numbers include: {19, 41, 73, 89, 113, 149, 167, 193, 223, 229, 269, 293, 337, 347, 367, 383, 521, ...}. Primes which are sums of three positive heptagonal numbers include: {3, 37, 43, 53, 59, 83, 89, 107, 131, 137, 149, 163, 167, 173, 191, 197, 211, 227, 241, 251, 257, 263, 271, ...}. Primes which are sums of four positive heptagonal numbers include: {37, 43, 61, 71, 97, 101, 103, 107, 109, 113, 127, 149, 151, 167, 181, 191, 197, 199, 211, 223, 229, 239, 251, ...}.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A000566, A000040, A000326, A003679, A064826, A117065.

Module[{upto=150,max},max=Ceiling[(3+Sqrt[9+40upto])/10];Select[Total/@
Tuples[PolygonalNumber[7,Range[max]],4]//Union,#<=upto&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 15 2016 *)

{a(n)} = {A000566} + {A000566} + {A000566} + {A000566} = {a*(5*a-3)/2 + b*(5*b-3)/2 + c*(5*c-3)/2 + d*(5*d-3)/2 such that every term is positive}.

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

Jonathan Vos Post, Sep 23 2010

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

Charles R Greathouse IV, Table of n, a(n) for n = 1..1671

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

cp's OEIS Frontend

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

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A100878 Smallest number of pentagonal numbers which sum to n.

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A117065 Primes that are not the sum of 3 pentagonal numbers.

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A280718 Expansion of (Sum_{k>=0} x^(k*(3*k-1)/2))^5.

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A117104 Sum of two positive heptagonal numbers A000566.

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A117105 Numbers that are the sum of three positive heptagonal numbers (A000566) in at least one way.

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A133929 Positive integers that cannot be expressed using four pentagonal numbers.

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A355660 Numbers m such that the smallest number of pentagonal numbers (A000326) which sum to m is exactly 4.

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A280718 Expansion of (Sum_{k>=0} x^(k(3k-1)/2))^5.