A355717 -id:A355717 - cp's OEIS frontend

A100878 Smallest number of pentagonal numbers which sum to n.

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

A355774 An extension of the generalized pentagonal numbers such that every positive integer can be represented as the sum of at most two terms of the sequence.

Original entry on oeis.org

0, 1, 2, 5, 7, 11, 12, 15, 21, 22, 25, 26, 35, 39, 40, 49, 51, 57, 67, 70, 77, 87, 92, 100, 117, 120, 123, 126, 145, 153, 155, 173, 176, 182, 186, 187, 205, 210, 214, 222, 228, 241, 247, 251, 260, 283, 287, 301, 319, 330, 345, 376, 382, 392, 425, 435, 442, 448

Offset: 0

Peter Luschny, Jul 17 2022

nonn

The sequence is defined inductively. Starting from the empty sequence, the terms are added one after the other. A term is added if it is a generalized pentagonal number or if it cannot be represented as the sum of two preceding terms. Note that these exceptions form a proper subsequence of A093519.

Thus any positive number can be expressed as the sum of at most two positive terms by Euler's Pentagonal Number Theorem. Every pentagonal number and every generalized pentagonal number is in this sequence.

			32 = 7 + 25; 195 = 22 + 173.

Andreas Enge, William Hart and Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016.
Burkard Polster (Mathloger), The hardest 'What comes next?' (Euler's pentagonal formula), YouTube video, 2020.

Cf. A000326, A001318, A093519, A100878, A355717, A176747 (same construction with triangular numbers).

A355774_list := proc(upto) local P, k, issum, isgpn; P := [];
isgpn := k -> ormap(n -> 0 = 8*k-(n+irem(n,2))*(3*n+2-irem(n,2)), [$0..k]);
issum := k -> ormap(p -> member(k - p, P), P);
for k from 0 to upto do
    if isgpn(k) or not issum(k) then P := [op(P), k] fi od;
P end: print(A355774_list(448));

isgpn[k_] := AnyTrue[Range[0, k], 0 == 8*k-(#+Mod[#,2])*(3*#+2-Mod[#,2])&];
issum[k_] := AnyTrue[P, MemberQ[P, k-#]&];
P = {};
For[k = 0, k <= 448, k++, If[isgpn[k] || !issum[k], AppendTo[P, k]]];
P (* Jean-François Alcover, Mar 07 2024, after Peter Luschny *)

def A355774_list(upto: int) -> list[int]:
    P: list[int] = []
    for k in range(upto + 1):
        if any(
            k == ((n + n % 2) * (3 * n + 2 - n % 2)) >> 3
            for n in range(k + 1)
        ) or not any([(k - p) in P for p in P]):
            P.append(k)
    return P
print(A355774_list(448))

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A100878 Smallest number of pentagonal numbers which sum to n.

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