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

Author

Yifan Xie, Apr 05 2023

Keywords

• nonn
• easy

Comments

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

Examples

			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.
		

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

Crossrefs

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

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

Programs

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)));

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