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%I #13 Feb 10 2021 09:38:34
%S 13,15,17,25,26,29,30,34,35,37,39,41,45,50,51,52,53,55,58,60,61,65,68,
%T 70,73,74,75,78,82,85,87,89,90,91,95,97,100,101,102,104,105,106,109,
%U 110,111,113,115,116,117,119,120,122,123,125,130,135,136,137
%N Numbers k such that k^2 is the sum of m nonzero squares for all 1 <= m <= k^2 - 14.
%C Numbers k such that k^2 is in A018820. Note that k^2 is never the sum of k^2 - 13 positive squares.
%C A square k^2 is the sum of m positive squares for all 1 <= m <= k^2 - 14 if k^2 is the sum of 2 and 3 positive squares (see A309778 and proof in Kuczma).
%C Intersection of A009003 and A005767. Also A009003 \ A020714.
%C Numbers k not of the form 5*2^e such that k has at least one prime factor congruent to 1 modulo 4.
%C Has density 1 over all positive integers.
%D Marcin E. Kuczma, International Mathematical Olympiads, 1986-1999, The Mathematical Association of America, 2003, pages 76-79.
%H Jianing Song, <a href="/A341329/b341329.txt">Table of n, a(n) for n = 1..10000</a>
%e 13 is a term: 169 = 13^2 = 5^2 + 12^2 = 3^2 + 4^2 + 12^2 = 11^2 + 4^2 + 4^2 + 4^2 = 6^2 + 6^2 + 6^2 + 6^2 + 5^2 = 6^2 + 6^2 + 6^2 + 6^2 + 4^2 + 3^2 = ... = 3^2 + 2^2 + 2^2 + 1^2 + 1^2 + ... + 1^2 (sum of 155 positive squares, with 152 (1^2)'s), but 169 cannot be represented as the sum of 156 positive squares.
%o (PARI) isA341329(n) = setsearch(Set(factor(n)[, 1]%4), 1) && !(n/5 == 2^valuation(n, 2))
%Y Cf. A018820, A309778, A009003, A005767, A020714.
%K nonn,easy
%O 1,1
%A _Jianing Song_, Feb 09 2021
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