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
Keywords
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.
Formula
a(n) = (A383682(n) - n) / 2.
Comments