A295819 Number of nonnegative solutions to (x,y) = 1 and x^2 + y^2 = n.

0, 2, 1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0

Offset: 0

Author

Seiichi Manyama, Nov 28 2017

Keywords

• nonn

Examples

			a(1) = 2;
(1,0) = 1 and 1^2 + 0^2 =  1.
(0,1) = 1 and 0^2 + 1^2 =  1.
a(2) = 1;
(1,1) = 1 and 1^2 + 1^2 =  2. ->  1^2 +  1^2 == 1^2 + 1 == 0 mod  2.
a(5) = 2;
(2,1) = 1 and 2^2 + 1^2 =  5. ->  2^2 +  1^2 == 2^2 + 1 == 0 mod  5.
(1,2) = 1 and 1^2 + 2^2 =  5. ->  3^2 +  6^2 == 3^2 + 1 == 0 mod  5.
a(10) = 2;
(3,1) = 1 and 3^2 + 1^2 = 10. ->  3^2 +  1^2 == 3^2 + 1 == 0 mod 10.
(1,3) = 1 and 1^2 + 3^2 = 10. ->  7^2 + 21^2 == 7^2 + 1 == 0 mod 10.
a(13) = 2;
(3,2) = 1 and 3^2 + 2^2 = 13. -> 21^2 + 14^2 == 8^2 + 1 == 0 mod 13.
(2,3) = 1 and 2^2 + 3^2 = 13. -> 18^2 + 27^2 == 5^2 + 1 == 0 mod 13.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A006278.

Similar sequences: A000010, A000925, A295820, A295848, A295976.

A000089 is essentially the same sequence.

Programs

Mathematica

a[n_] := Sum[j = Sqrt[n - i^2] // Floor; Boole[GCD[i, j] == 1 && i^2 + j^2 == n], {i, 0, Sqrt[n]}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 05 2018, after Andrew Howroyd *)

PARI

a(n) = {sum(i=0, sqrtint(n), my(j=sqrtint(n-i^2)); gcd(i,j)==1 && i^2+j^2==n)} \\ Andrew Howroyd, Dec 12 2017

Formula

a(n) = A000089(n) for n >= 2.

a(A006278(n)) = 2^n for n >= 1.