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A344810 Numbers that are the sum of six squares in six or more ways.

%I A344810 #14 May 10 2024 01:38:51
%S A344810 54,57,60,62,65,68,69,71,72,75,76,77,78,80,81,83,84,86,87,88,89,90,92,
%T A344810 93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,
%U A344810 112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127
%N A344810 Numbers that are the sum of six squares in six or more ways.
%H A344810 Sean A. Irvine, <a href="/A344810/b344810.txt">Table of n, a(n) for n = 1..1000</a>
%F A344810 Conjectures from _Chai Wah Wu_, Apr 25 2024: (Start)
%F A344810 a(n) = 2*a(n-1) - a(n-2) for n > 24.
%F A344810 G.f.: x*(-x^23 + x^22 - x^18 + x^17 - x^16 + x^15 - x^14 + x^13 - 2*x^10 + 2*x^9 - x^8 + x^7 - 2*x^6 + x^4 - x^3 - 51*x + 54)/(x - 1)^2. (End)
%e A344810 57 = 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 7^2
%e A344810    = 1^2 + 1^2 + 1^2 + 2^2 + 5^2 + 5^2
%e A344810    = 1^2 + 1^2 + 1^2 + 3^2 + 3^2 + 6^2
%e A344810    = 1^2 + 2^2 + 2^2 + 4^2 + 4^2 + 4^2
%e A344810    = 1^2 + 2^2 + 3^2 + 3^2 + 3^2 + 5^2
%e A344810    = 2^2 + 2^2 + 2^2 + 2^2 + 4^2 + 5^2
%e A344810 so 57 is a term.
%o A344810 (Python)
%o A344810 from itertools import combinations_with_replacement as cwr
%o A344810 from collections import defaultdict
%o A344810 keep = defaultdict(lambda: 0)
%o A344810 power_terms = [x**2 for x in range(1, 1000)]
%o A344810 for pos in cwr(power_terms, 6):
%o A344810     tot = sum(pos)
%o A344810     keep[tot] += 1
%o A344810     rets = sorted([k for k, v in keep.items() if v >= 6])
%o A344810     for x in range(len(rets)):
%o A344810         print(rets[x])
%Y A344810 Cf. A344799, A344809, A344811, A345483, A345515.
%K A344810 nonn
%O A344810 1,1
%A A344810 _David Consiglio, Jr._, Jun 20 2021