A295819 - OEIS

%I #63 Oct 28 2018 04:50:50

%S 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,

%T 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,

%U 0,0,0,0,0,2,2,0,0,0,0,0,0,0,2,0,0,4,0,0

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

%H Seiichi Manyama, <a href="/A295819/b295819.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%e a(1) = 2;

%e (1,0) = 1 and 1^2 + 0^2 =  1.

%e (0,1) = 1 and 0^2 + 1^2 =  1.

%e a(2) = 1;

%e (1,1) = 1 and 1^2 + 1^2 =  2. ->  1^2 +  1^2 == 1^2 + 1 == 0 mod  2.

%e a(5) = 2;

%e (2,1) = 1 and 2^2 + 1^2 =  5. ->  2^2 +  1^2 == 2^2 + 1 == 0 mod  5.

%e (1,2) = 1 and 1^2 + 2^2 =  5. ->  3^2 +  6^2 == 3^2 + 1 == 0 mod  5.

%e a(10) = 2;

%e (3,1) = 1 and 3^2 + 1^2 = 10. ->  3^2 +  1^2 == 3^2 + 1 == 0 mod 10.

%e (1,3) = 1 and 1^2 + 3^2 = 10. ->  7^2 + 21^2 == 7^2 + 1 == 0 mod 10.

%e a(13) = 2;

%e (3,2) = 1 and 3^2 + 2^2 = 13. -> 21^2 + 14^2 == 8^2 + 1 == 0 mod 13.

%e (2,3) = 1 and 2^2 + 3^2 = 13. -> 18^2 + 27^2 == 5^2 + 1 == 0 mod 13.

%t a[n_] := Sum[j = Sqrt[n - i^2] // Floor; Boole[GCD[i, j] == 1 && i^2 + j^2 == n], {i, 0, Sqrt[n]}];

%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Jul 05 2018, after _Andrew Howroyd_ *)

%o (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

%Y Cf. A006278.

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

%Y A000089 is essentially the same sequence.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 28 2017