A360530 a(n) is the smallest positive integer k such that n can be expressed as the arithmetic mean of k nonzero squares.

1, 3, 3, 1, 2, 3, 3, 3, 1, 2, 3, 3, 2, 3, 3, 1, 2, 3, 3, 2, 4, 3, 3, 3, 1, 2, 3, 3, 2, 3, 3, 3, 3, 2, 3, 1, 2, 3, 3, 2, 2, 3, 3, 3, 2, 3, 3, 3, 1, 2, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 1, 2, 3, 3, 2, 4, 3, 3, 3, 2, 2, 3, 3, 4, 3, 3, 2, 1, 2, 3, 4, 2, 3, 3

Offset

1,2

Comments

a(n) is the smallest number k such that n*k can be expressed as the sum of k nonzero squares.

References

J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, p. 140.

Links

Yifan Xie, Table of n, a(n) for n = 1..10000

Wikipedia, Lagrange's four-square theorem

Index entries for sequences related to sums of squares

Formula

a(n) <= 4. Proof: With Lagrange's four-square theorem, if 4*n is not the sum of 4 positive squares (see A000534), then it is easy to express 3*n as the sum of 3 positive squares. - Yifan Xie and Thomas Scheuerle, Apr 29 2023

Example

For n = 2, if k = 1, 2*1 = 2 is a nonsquare; if k = 2, 2*2 = 4 cannot be expressed as the sum of 2 nonzero squares; if k = 3, 2*3 = 6 = 2^2+1^2+1^2, so a(2) = 3.

Prog

(PARI)
findsquare(k, m) = if(k == 1, issquare(m), for(j=1, m, if(j*j+k > m, return(0), if(findsquare(k-1, m-j*j), return(1)))));
a(n) = for(t = 1, n+1, if(findsquare(t, n*t), return(t)));

Crossrefs

Cf. A000290, A000378, A000404, A000408, A000414, A000534, A047700.

Cf. A362068 (allows zeros), A362110 (distinct).

Sequence in context: A225331 A004550 A096836 * A096995 A255941 A010264

Adjacent sequences: A360527 A360528 A360529 * A360531 A360532 A360533

Keyword

nonn,easy

Author

Yifan Xie, Apr 05 2023

Status

approved