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

169, 225, 289, 625, 676, 841, 900, 1156, 1225, 1369, 1521, 1681, 2025, 2500, 2601, 2704, 2809, 3025, 3364, 3600, 3721, 4225, 4624, 4900, 5329, 5476, 5625, 6084, 6724, 7225, 7569, 7921, 8100, 8281, 9025, 9409, 10000, 10201, 10404, 10816, 11025, 11236

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

Intersection of A000290, A000404 and A000408. - Zak Seidov, Nov 12 2013

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

Note that k is never the sum of k - 13 positive squares. - Jianing Song, Feb 09 2021

Examples

			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
		

References

• Marcin E. Kuczma, International Mathematical Olympiads, 1986-1999, The Mathematical Association of America, 2003, pages 76-79.

Links

• Zak Seidov and Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..100 from Zak Seidov)
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A000290, A000404, A000408, A309778, A341329.

Programs

PARI

isA018820(n) = issquare(n) && isA341329(sqrtint(n)) \\ Jianing Song, Feb 09 2021, see A341329 for its program

Formula

a(n) = A341329(n)^2. - Jianing Song, Feb 09 2021