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%I #14 May 10 2024 01:40:15
%S 70,79,82,85,87,88,90,91,93,94,95,96,97,98,99,100,101,102,103,104,105,
%T 106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,
%U 123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139
%N Numbers that are the sum of seven squares in ten or more ways.
%H Sean A. Irvine, <a href="/A345487/b345487.txt">Table of n, a(n) for n = 1..1000</a>
%F Conjectures from _Chai Wah Wu_, Jan 05 2024: (Start)
%F a(n) = 2*a(n-1) - a(n-2) for n > 10.
%F G.f.: x*(-x^9 + x^8 - x^7 + x^6 - x^5 - x^4 - 6*x^2 - 61*x + 70)/(x - 1)^2. (End)
%e 79 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 5^2 + 7^2
%e = 1^2 + 1^2 + 1^2 + 1^2 + 5^2 + 5^2 + 5^2
%e = 1^2 + 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 8^2
%e = 1^2 + 1^2 + 1^2 + 3^2 + 3^2 + 3^2 + 7^2
%e = 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 4^2 + 7^2
%e = 1^2 + 1^2 + 2^2 + 4^2 + 4^2 + 4^2 + 5^2
%e = 1^2 + 1^2 + 3^2 + 3^2 + 3^2 + 5^2 + 5^2
%e = 1^2 + 2^2 + 2^2 + 2^2 + 4^2 + 5^2 + 5^2
%e = 1^2 + 2^2 + 2^2 + 3^2 + 3^2 + 4^2 + 6^2
%e = 2^2 + 3^2 + 3^2 + 3^2 + 4^2 + 4^2 + 4^2
%e = 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 5^2
%e so 79 is a term.
%o (Python)
%o from itertools import combinations_with_replacement as cwr
%o from collections import defaultdict
%o keep = defaultdict(lambda: 0)
%o power_terms = [x**2 for x in range(1, 1000)]
%o for pos in cwr(power_terms, 7):
%o tot = sum(pos)
%o keep[tot] += 1
%o rets = sorted([k for k, v in keep.items() if v >= 10])
%o for x in range(len(rets)):
%o print(rets[x])
%Y Cf. A345477, A345486, A345497, A345506.
%K nonn
%O 1,1
%A _David Consiglio, Jr._, Jun 20 2021
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