A383682 The largest nonnegative integer value of j for which each integer n, n+2, ..., j-4, j-2, j can be written as the sum of the squares of the elements of a partition of n.

1, 4, 5, 10, 13, 14, 21, 34, 35, 46, 61, 62, 77, 78, 95, 114, 121, 142, 165, 190, 225, 246, 277, 290, 345, 358, 359, 396, 435, 446, 487, 530, 575, 622, 679, 722, 783, 790, 791, 846, 903, 1022, 1085, 1086, 1151, 1230, 1287, 1358, 1373, 1374, 1521, 1522, 1599

Offset

1,2

Links

Table of n, a(n) for n=1..53.

Bradley K. Moon and Noah A. Rosenberg, Integer Sequences for Diversity Statistics, J. Int. Seq. 29(1) (2026), 26.1.5. See p. 9.

Bruce Reznick, The sum of the squares of the parts of a partition, and some related questions, J. Number Theory 33 (1989), 199-208.

Peter Winkler, Mean distance in a tree, Discr. Appl. Math. (1990), 179-185.

Formula

a(n) ~ n^2-2*sqrt(2)*n^(3/2)+O(n^(5/4)) (Reznick 1989, p. 201).

Example

Consider n=3: 3 and 5 can be written as sums of squares of partitions of 3, as 3=1^2+1^2+1^2 and 5=2^2+1^2, but 7 cannot be written as a sum of squares of a partition of 3, so a(3)=5.
Consider n=4: 4, 6, 8, and 10 can be written as sums of squares of partitions of 4, as 4=1^2+1^2+1^2+1^2, 6=2^2+1^2+1^2, 8=2^2+2^2, and 10=3^2+1^2, but 12 cannot be written as a sum of squares of a partition of 4, so a(4)=10.

Crossrefs

Cf. A381811.

Sequence in context: A094415 A114517 A394927 * A283246 A236283 A394391

Adjacent sequences: A383679 A383680 A383681 * A383683 A383684 A383685

Keyword

nonn

Author

Noah A Rosenberg, May 05 2025

Status

approved