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
Keywords
Examples
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.
Links
- B. Reznick, The sum of the squares of the parts of a partition, and some related questions, J. Number Theory 33 (1989), 199-208.
- P. Winkler, Mean distance in a tree, Discr. Appl. Math. (1990), 179-185.
Crossrefs
Cf. A381811.
Formula
a(n) ~ n^2-2*sqrt(2)*n^(3/2)+O(n^(5/4)) (Reznick 1989, p. 201).