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

1,5

Links

Table of n, a(n) for n=1..99.

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

Example

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.

Crossrefs

Cf. A360530 (allows repeated squares).

Cf. A001422, A033461, A129210, A132777, A248509.

Sequence in context: A088673 A339662 A336316 * A236138 A363930 A361755

Adjacent sequences: A362107 A362108 A362109 * A362111 A362112 A362113

Keyword

nonn

Author

Yifan Xie, Apr 16 2023

Extensions

Name qualified and other edits by Peter Munn, Apr 21 2023

Status

approved