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A345484 Numbers that are the sum of seven squares in seven or more ways.

%I A345484 #10 Apr 26 2024 05:48:49
%S A345484 55,58,61,63,64,66,69,70,72,73,74,75,76,77,78,79,80,81,82,84,85,86,87,
%T A345484 88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,
%U A345484 108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123
%N A345484 Numbers that are the sum of seven squares in seven or more ways.
%H A345484 Sean A. Irvine, <a href="/A345484/b345484.txt">Table of n, a(n) for n = 1..1000</a>
%F A345484 Conjectures from _Chai Wah Wu_, Apr 25 2024: (Start)
%F A345484 a(n) = 2*a(n-1) - a(n-2) for n > 21.
%F A345484 G.f.: x*(-x^20 + x^19 - x^9 + x^8 - 2*x^7 + x^6 + x^5 - x^4 - x^3 - 52*x + 55)/(x - 1)^2. (End)
%e A345484 58 is a term because 58 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 7^2 = 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 5^2 + 5^2 = 1^2 + 1^2 + 1^2 + 1^2 + 3^2 + 3^2 + 6^2 = 1^2 + 1^2 + 2^2 + 2^2 + 4^2 + 4^2 + 4^2 = 1^2 + 1^2 + 2^2 + 3^2 + 3^2 + 3^2 + 5^2 = 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 4^2 + 5^2 = 2^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2.
%o A345484 (Python)
%o A345484 from itertools import combinations_with_replacement as cwr
%o A345484 from collections import defaultdict
%o A345484 keep = defaultdict(lambda: 0)
%o A345484 power_terms = [x**2 for x in range(1, 1000)]
%o A345484 for pos in cwr(power_terms, 7):
%o A345484     tot = sum(pos)
%o A345484     keep[tot] += 1
%o A345484     rets = sorted([k for k, v in keep.items() if v >= 7])
%o A345484     for x in range(len(rets)):
%o A345484         print(rets[x])
%Y A345484 Cf. A344811, A345483, A345485, A345494, A345525.
%K A345484 nonn
%O A345484 1,1
%A A345484 _David Consiglio, Jr._, Jun 20 2021