A151925 Write n as a sum of positive squares a^2+b^2+c^2+... with gcd(a,b,...) = 1; a(n) = minimal number of squares needed.

1, 2, 3, 4, 2, 3, 4, 5, 3, 2, 3, 4, 2, 3, 4, 5, 2, 3, 3, 4, 3, 3, 4, 5, 2, 2, 3, 4, 2, 3, 4, 5, 3, 2, 3, 4, 2, 3, 4, 5, 2, 3, 3, 4, 3, 3, 4, 5, 3, 2, 3, 4, 2, 3, 4, 5, 3, 2, 3, 4, 2, 3, 4, 5, 2, 3, 3, 4, 3, 3, 4, 5, 2, 2, 3, 4, 3, 3, 4, 5, 3, 2, 3, 4, 2, 3, 4, 5, 2, 3, 3, 4, 3, 3, 4, 5, 2, 3, 3

Offset: 1

Author

N. J. A. Sloane and Vinay Vaishampayan, Aug 06 2009, Aug 07 2009

Keywords

• nonn

Comments

Similar to A002828, but only now primitive representations are allowed.

Of course a(n) >= A002828(n).

From Lagrange's theorem, a(n) <= 5 (see also Estermann, Grosswald, Th. 3, p. 176).

Furthermore, it appears (and should be easy to prove) that:

a(n) = 1 iff n=1

a(n) = 2 iff n in A008784\{1}

a(n) = 3 iff n in A151926

a(n) = 4 iff n == 4 or 7 mod 8

a(n) = 5 iff n == 0 mod 8

Examples

			..... n .. a(n) ..<- Numbers when squared add to n ->
-----------------------------------------------------
......1......1......1
......2......2......1......1
......3......3......1......1......1
......4......4......1......1......1......1
......5......2......1......2
......6......3......1......1......2
......7......4......1......1......1......2
......8......5......1......1......1......1......2
......9......3......1......2......2
.....10......2......1......3
.....11......3......1......1......3
.....12......4......1......1......1......3
.....13......2......2......3
.....14......3......1......2......3
.....15......4......1......1......2......3
.....16......5......1......1......1......2......3
.....17......2......1......4
.....18......3......1......1......4
.....19......3......1......3......3
.....20......4......1......1......3......3
		

References

• Estermann, T., On the representations of a number as a sum of squares, Acta Arith., 45 (1937), 93-125.
• E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..1000
• N. J. A. Sloane, Fortran program