A256452 Number of integer solutions to n^2 = x^2 + y^2 with x>0, y>=0.

1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 3, 1, 1, 3, 1, 1, 1, 1, 5, 3, 1, 1, 3, 3, 1, 1, 1, 3, 3, 1, 3, 1, 3, 3, 3, 1, 1, 1, 3, 1, 1, 1, 1, 5, 3, 3, 3, 1, 3, 1, 1, 3, 1, 3, 3, 1, 1, 1, 9, 1, 1, 3, 1, 3, 1, 1, 3, 3, 5, 1, 1, 3, 1, 3, 1, 3, 1, 1, 9, 1, 3

Offset: 1

Author

Michael Somos, Mar 29 2015

Keywords

• nonn
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A046080, A046109, A002654.

Programs

Maple

a:= n-> add(`if`(d::odd, (-1)^((d-1)/2), 0), d=numtheory[divisors](n^2)): seq(a(n), n=1..100);  # Ridouane Oudra, Aug 18 2024

Mathematica

a[ n_] := Sum[ Mod[ Length@Divisors[n^2 - k^2], 2], {k, n}];
a[ n_] := Length @ FindInstance[ n^2 == x^2 + y^2 && x > 0 && y >= 0, {x, y}, Integers, 10^9]; (* Michael Somos, Aug 15 2016 *)
f[p_, e_] := If[Mod[p, 4] == 1, 2*e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 12 2020 *)

PARI

{a(n) = sum(k=1, n, issquare(n^2 - k^2))};

Formula

Multiplicative with a(p^e) = 2*e + 1 if p == 1 (mod 4), otherwise a(p^e) = 1.

a(n) = 1 + 2*A046080(n) if n>0.

a(n) = A046109(n)/4 for n > 0. - Hugo Pfoertner, Sep 21 2023

a(n) = A002654(n^2). - Ridouane Oudra, Aug 18 2024