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A018820 Numbers k that are the sum of m nonzero squares for all 1 <= m <= k - 14.

%I A018820 #22 Feb 12 2021 17:51:09
%S A018820 169,225,289,625,676,841,900,1156,1225,1369,1521,1681,2025,2500,2601,
%T A018820 2704,2809,3025,3364,3600,3721,4225,4624,4900,5329,5476,5625,6084,
%U A018820 6724,7225,7569,7921,8100,8281,9025,9409,10000,10201,10404,10816,11025,11236
%N A018820 Numbers k that are the sum of m nonzero squares for all 1 <= m <= k - 14.
%C A018820 Intersection of A000290, A000404 and A000408. - _Zak Seidov_, Nov 12 2013
%C A018820 A square k^2 is the sum of m positive squares for all 1 <= m <= k^2 - 14 iff k^2 is the sum of 2 and 3 positive squares (see A309778 and proof in Kuczma). - _Bernard Schott_, Aug 17 2019
%C A018820 Note that k is never the sum of k - 13 positive squares. - _Jianing Song_, Feb 09 2021
%D A018820 Marcin E. Kuczma, International Mathematical Olympiads, 1986-1999, The Mathematical Association of America, 2003, pages 76-79.
%H A018820 Zak Seidov and Chai Wah Wu, <a href="/A018820/b018820.txt">Table of n, a(n) for n = 1..10000</a> (n = 1..100 from Zak Seidov)
%H A018820 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A018820 a(n) = A341329(n)^2. - _Jianing Song_, Feb 09 2021
%e A018820 169 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. - _Jianing Song_, Feb 09 2021
%o A018820 (PARI) isA018820(n) = issquare(n) && isA341329(sqrtint(n)) \\ _Jianing Song_, Feb 09 2021, see A341329 for its program
%Y A018820 Cf. A000290, A000404, A000408, A309778, A341329.
%K A018820 nonn
%O A018820 1,1
%A A018820 _David W. Wilson_