A117111 Sum of four positive heptagonal numbers A000566.

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

Author

Jonathan Vos Post, Apr 18 2006

Keywords

• easy
• nonn

Comments

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, ...}.

Links

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

Crossrefs

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

Programs

Mathematica

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 *)

Formula

{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}.