|
| |
|
|
A074590
|
|
Number of primitive solutions to n = x^2 + y^2 + z^2 (i.e. with gcd(x,y,z) = 1).
|
|
2
| |
|
|
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
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
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
|
|
|
FORMULA
| n is representable as the sum of 3 squares iff 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)
|
|
|
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}](* From Jean-François Alcover, Jan 13 2012 *)
|
|
|
CROSSREFS
| Cf. A005875 (all solutions).
Sequence in context: A175375 A175365 A029769 * A105730 A005875 A028659
Adjacent sequences: A074587 A074588 A074589 * A074591 A074592 A074593
|
|
|
KEYWORD
| nonn,easy,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Dec 03 2002
|
|
|
EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 04 2002
|
| |
|
|
