A362110 a(n) is the smallest integer k such that n can be expressed as the arithmetic mean of k distinct nonzero squares, or 0 if no such k exists.
1, 0, 0, 1, 2, 0, 3, 0, 1, 2, 5, 0, 2, 3, 3, 1, 2, 3, 5, 2, 4, 3, 3, 5, 1, 2, 3, 3, 2, 3, 3, 5, 5, 2, 3, 1, 2, 3, 3, 2, 2, 3, 3, 5, 2, 3, 3, 5, 1, 2, 3, 2, 2, 3, 3, 3, 3, 2, 4, 3, 2, 3, 3, 1, 2, 3, 3, 2, 4, 3, 3, 3, 2, 2, 3, 5, 4, 3, 3, 2, 1, 2, 3, 4, 2, 3, 3, 3, 2, 2, 3, 3, 4, 3, 3, 5, 2, 3, 3
Offset: 1
Keywords
Examples
For n = 2, if k = 1, 2*1 = 2 is not a square; but, from the upper bound formula, (k + 1) * (2*k + 1) <= 12, so k <= 1. So, a satisfactory k does not exist; hence a(2) = 0.
Formula
Upper bound: (a(n) + 1) * (2*a(n) + 1) <= 6*n. Proof: Because (Sum_{m=1..k} (i_m)^2)/k = n, n*k = Sum_{m=1..k} (i_m)^2. Since each i_m is distinct, n*k >= Sum_{m=1..k} m^2 = k * (k + 1) * (2*k + 1)/6, hence (k + 1) * (2*k + 1) <= 6*n.
a(A132777(n)) = 2. - Thomas Scheuerle, Apr 16 2023
Extensions
Name qualified and other edits by Peter Munn, Apr 21 2023