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A345477 Numbers that are the sum of six squares in ten or more ways.

%I A345477 #19 May 10 2024 01:39:37
%S A345477 81,84,86,89,92,93,95,100,101,102,104,105,107,108,110,111,113,114,116,
%T A345477 117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,
%U A345477 134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150
%N A345477 Numbers that are the sum of six squares in ten or more ways.
%H A345477 David A. Corneth, <a href="/A345477/b345477.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Sean A. Irvine)
%F A345477 Conjectures from _Chai Wah Wu_, Jan 05 2024: (Start)
%F A345477 a(n) = 2*a(n-1) - a(n-2) for n > 20.
%F A345477 G.f.: x*(-x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - 4*x^8 + 3*x^7 + x^6 - 2*x^5 + x^3 - x^2 - 78*x + 81)/(x - 1)^2. (End)
%e A345477 84 = 1^2 + 1^2 + 1^2 + 1^2 + 4^2 + 8^2
%e A345477    = 1^2 + 1^2 + 1^2 + 3^2 + 6^2 + 6^2
%e A345477    = 1^2 + 1^2 + 1^2 + 4^2 + 4^2 + 7^2
%e A345477    = 1^2 + 1^2 + 2^2 + 2^2 + 5^2 + 7^2
%e A345477    = 1^2 + 1^2 + 4^2 + 4^2 + 5^2 + 5^2
%e A345477    = 1^2 + 2^2 + 2^2 + 5^2 + 5^2 + 5^2
%e A345477    = 1^2 + 2^2 + 3^2 + 3^2 + 5^2 + 6^2
%e A345477    = 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 8^2
%e A345477    = 2^2 + 2^2 + 3^2 + 3^2 + 3^2 + 7^2
%e A345477    = 2^2 + 4^2 + 4^2 + 4^2 + 4^2 + 4^2
%e A345477    = 3^2 + 3^2 + 3^2 + 4^2 + 4^2 + 5^2
%e A345477 so 84 is a term.
%o A345477 (Python)
%o A345477 from itertools import combinations_with_replacement as cwr
%o A345477 from collections import defaultdict
%o A345477 keep = defaultdict(lambda: 0)
%o A345477 power_terms = [x**2 for x in range(1, 1000)]
%o A345477 for pos in cwr(power_terms, 6):
%o A345477     tot = sum(pos)
%o A345477     keep[tot] += 1
%o A345477     rets = sorted([k for k, v in keep.items() if v >= 10])
%o A345477     for x in range(len(rets)):
%o A345477         print(rets[x])
%Y A345477 Cf. A025430, A344803, A345476, A345487, A345519.
%K A345477 nonn
%O A345477 1,1
%A A345477 _David Consiglio, Jr._, Jun 20 2021