A117111 - OEIS

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

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OFFSET

1,1

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

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}.

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 *)

CROSSREFS

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

Sequence in context: A223961 A224391 A224024 * A310508 A310509 A310510

Adjacent sequences: A117108 A117109 A117110 * A117112 A117113 A117114

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Apr 18 2006

STATUS

approved