A381811 The largest nonnegative integer j for which each integer n,n+2,...,n+2j can be written as the sum of the squares for some partition of n.

0, 1, 1, 3, 4, 4, 7, 13, 13, 18, 25, 25, 32, 32, 40, 49, 52, 62, 73, 85, 102, 112, 127, 133, 160, 166, 166, 184, 203, 208, 228, 249, 271, 294, 322, 343, 373, 376, 376, 403, 431, 490, 521, 521, 553, 592, 620, 655, 662, 662, 735, 735, 773, 812, 852, 893, 901, 943, 986

Offset: 1

Author

Noah A Rosenberg, May 05 2025

Keywords

• nonn

Comments

a(n) has an asymptotic equivalence with (1/2)*n^2-sqrt(2)*n^(3/2)+O(n^(5/4)) (Reznick 1989, p. 201).

Examples

			a(3) = 1, because n, n+2 (3 and 5) can be written as the sum of the squares for some partition of n; 3=1^2+1^2+1^2 and 5=2^2+1^2. However, 7 cannot be written as the sum of squares of the parts of a partition of 3, so a(3) = 1.
a(4) = 3, because n, n+2, n+4 and n+6 (4, 6, 8 and 10) can be written as the sum of the squares for some partition of n; 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. However, 12 cannot be written as the sum of squares of the parts of a partition of 4, so a(4) = 3.
		

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. A069999 (a(n) provides a lower bound for A069999(n)).

Cf. A000041, A383682.

Formula

a(n) = (A383682(n) - n) / 2.