A385963 a(n) is the maximum number of distinct positive integers whose sum of squares is equal to n^2.

0, 1, 1, 1, 1, 2, 1, 3, 1, 4, 5, 5, 5, 6, 7, 7, 7, 8, 8, 9, 9, 9, 9, 11, 10, 11, 11, 11, 11, 11, 13, 12, 12, 13, 14, 13, 14, 14, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 22, 24, 24, 24, 24, 24, 25

Offset: 0

Author

Gonzalo Martínez, Jul 13 2025

Keywords

• nonn

Comments

An upper bound is a(n) <= r for the largest r with 1^2 + ... + r^2 <= n^2, and with equality only at n = 0,1,24, the latter being a(70) = 24 (see comments A001032).

Examples

			For n = 11, there are A030273(11) = 4 partitions of 11^2 into distinct squares: {11^2}, {2^2, 6^2, 9^2}, {1^2, 2^2, 4^2, 10^2}, {1^2, 2^2, 4^2, 6^2, 8^2}, where the largest cardinality of these sets is 5. Therefore, a(11) = 5.
		

Links

• David A. Corneth, Table of n, a(n) for n = 0..10000
• David A. Corneth, n, a(n), one tuple of size a(n) where sum of elements of tuples is n^2

Crossrefs

Cf. A030273, A001032.

Programs

PARI

a(n)=poldegree(polcoef(prod(k=1, n, 1 + y*x^(k^2), 1 + O(x^(n^2+1))), n^2)) \\ Andrew Howroyd, Jul 13 2025

Extensions

More terms from Andrew Howroyd, Jul 13 2025