A341329 Numbers k such that k^2 is the sum of m nonzero squares for all 1 <= m <= k^2 - 14.

13, 15, 17, 25, 26, 29, 30, 34, 35, 37, 39, 41, 45, 50, 51, 52, 53, 55, 58, 60, 61, 65, 68, 70, 73, 74, 75, 78, 82, 85, 87, 89, 90, 91, 95, 97, 100, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120, 122, 123, 125, 130, 135, 136, 137

Offset

1,1

Comments

Numbers k such that k^2 is in A018820. Note that k^2 is never the sum of k^2 - 13 positive squares.

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).

Intersection of A009003 and A005767. Also A009003 \ A020714.

Numbers k not of the form 5*2^e such that k has at least one prime factor congruent to 1 modulo 4.

Has density 1 over all positive integers.

References

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

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Example

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.

Prog

(PARI) isA341329(n) = setsearch(Set(factor(n)[, 1]%4), 1) && !(n/5 == 2^valuation(n, 2))

Crossrefs

Cf. A018820, A309778, A009003, A005767, A020714.

Sequence in context: A120129 A350867 A354745 * A109019 A068893 A079829

Adjacent sequences: A341326 A341327 A341328 * A341330 A341331 A341332

Keyword

nonn,easy

Author

Jianing Song, Feb 09 2021

Status

approved