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%I #8 Oct 15 2016 11:28:05
%S 4,10,16,21,22,27,28,33,37,38,39,43,44,49,50,54,55,58,60,61,64,66,70,
%T 71,72,75,76,77,81,82,84,87,88,90,91,92,93,96,97,98,101,102,103,104,
%U 107,108,109,112,113,114,115,117,118,120,121,123,124,125,127,129,130,132
%N Sum of four positive heptagonal numbers A000566.
%C 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, ...}.
%H Harvey P. Dale, <a href="/A117111/b117111.txt">Table of n, a(n) for n = 1..2000</a>
%F {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}.
%t Module[{upto=150,max},max=Ceiling[(3+Sqrt[9+40upto])/10];Select[Total/@
%t Tuples[PolygonalNumber[7,Range[max]],4]//Union,#<=upto&]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Oct 15 2016 *)
%Y Cf. A000566, A000040, A000326, A003679, A064826, A117065.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Apr 18 2006
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