A256452 - OEIS

%I #20 Aug 18 2024 11:45:55

%S 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,

%T 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,

%U 1,3,1,1,3,3,5,1,1,3,1,3,1,3,1,1,9,1,3

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

%H Amiram Eldar, <a href="/A256452/b256452.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F a(n) = A046109(n)/4 for n > 0. - _Hugo Pfoertner_, Sep 21 2023

%F a(n) = A002654(n^2). - _Ridouane Oudra_, Aug 18 2024

%p 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

%t a[ n_] := Sum[ Mod[ Length@Divisors[n^2 - k^2], 2], {k, n}];

%t a[ n_] := Length @ FindInstance[ n^2 == x^2 + y^2 && x > 0 && y >= 0, {x, y}, Integers, 10^9]; (* _Michael Somos_, Aug 15 2016 *)

%t 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 *)

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

%Y Cf. A046080, A046109, A002654.

%K nonn,mult

%O 1,5

%A _Michael Somos_, Mar 29 2015