A074590 Number of primitive solutions to n = x^2 + y^2 + z^2 (i.e., with gcd(x,y,z) = 1).

1, 6, 12, 8, 0, 24, 24, 0, 0, 24, 24, 24, 0, 24, 48, 0, 0, 48, 24, 24, 0, 48, 24, 0, 0, 24, 72, 24, 0, 72, 48, 0, 0, 48, 48, 48, 0, 24, 72, 0, 0, 96, 48, 24, 0, 48, 48, 0, 0, 48, 72, 48, 0, 72, 72, 0, 0, 48, 24, 72, 0, 72, 96, 0, 0, 96, 96, 24, 0, 96, 48, 0, 0, 48, 120

Offset: 0

Author

N. J. A. Sloane, Dec 03 2002

Keywords

• nonn
• easy
• nice

Examples

			G.f. = 1 + 6*x + 12*x^2 + 8*x^3 + 24*x^5 + 24*x^6 + 24*x^9 + 24*x^10 + 24*x^11 + ...
		

References

• See A005875 for references.
• E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 54.

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A005875 (all solutions).

Programs

Mathematica

a[n_] := (r = Reduce[ GCD[x, y, z] == 1 && n == x^2 + y^2 + z^2, {x, y, z}, Integers]; If[ r === False, 0, Length[ {ToRules[r]} ] ] ); a[0] = 1; Table[ a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 13 2012 *)
a[ n_] := If[ n < 1, Boole[n == 0], Length @ Select[ {x, y, z} /. FindInstance[ n == x^2 + y^2 + z^2, {x, y, z}, Integers, 10^9], 1 == GCD @@ # &]]; (* Michael Somos, May 21 2015 *)

Formula

n is representable as the sum of 3 squares if and only if n is not of the form 4^a (8k + 7) (cf. A000378).

A005875(n) = Sum_{d^2|n} a(n/d^2).

Let h = number of classes of primitive binary quadratic forms, corresponding to the discriminant D = -n if n = 3 (mod 8), D = -4n if n = 1, 2, 5, 6 (mod 8) and let d_1 = 1/2, d_3 = 1/3, d_n = 1 otherwise. Then a(n) = 12 h d_n, if n = 1, 2, 5, 6 (mod 8), 24 h d_n, if n = 3 (mod 8). (Grosswald)

Also, if n is squarefree and (r/n) is the Jacobi symbol, a(n) = 24 sum(r = 1, [n/4], (r/n)) if n = 1 (mod 4), 8 sum(r = 1, [n/2], (r/n)) if n = 3 (mod 8). (Grosswald)

Extensions

More terms from Vladeta Jovovic, Dec 04 2002