A235872 Number of solutions to the equation x^2=0 in the ring of Gaussian integers modulo n.

1, 2, 1, 4, 1, 2, 1, 8, 9, 2, 1, 4, 1, 2, 1, 16, 1, 18, 1, 4, 1, 2, 1, 8, 25, 2, 9, 4, 1, 2, 1, 32, 1, 2, 1, 36, 1, 2, 1, 8, 1, 2, 1, 4, 9, 2, 1, 16, 49, 50, 1, 4, 1, 18, 1, 8, 1, 2, 1, 4, 1, 2, 9, 64, 1, 2, 1, 4, 1, 2, 1, 72, 1, 2, 25, 4, 1, 2, 1, 16, 81, 2

Offset

1,2

Comments

Numbers of solutions to x^2 == y^2 (mod n), 2*x*y == 0 (mod n). - Andrew Howroyd, Aug 06 2018

Links

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(2^e) = 2^e, a(p^e) = p^(2*floor(e/2)). - Andrew Howroyd, Aug 06 2018

Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (2/21)*(3+sqrt(2))*zeta(3/2)/zeta(3) = 0.91363892007.... - Amiram Eldar, Nov 13 2022

Mathematica

invoG[n_] := invoG[n] = Sum[If[Mod[(x + I y)^2, n] == 0, 1, 0], {x, 0, n - 1}, {y, 0, n - 1}]; Table[invoG[n], {n, 1, 104}]
f[p_, e_] := p^(2*Floor[e/2]); f[2, e_] := 2^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 13 2022 *)

Prog

(PARI) a(n)={sum(i=0, n-1, sum(j=0, n-1, (i^2 - j^2)%n == 0 && 2*i*j%n == 0))} \\ Andrew Howroyd, Aug 06 2018
(PARI) a(n)={my(f=factor(n)); prod(i=1, #f~, my([p, e]=f[i, ]); p^if(p==2, e, e - e%2))} \\ Andrew Howroyd, Aug 06 2018

Crossrefs

Cf. A062803, A090699.

Sequence in context: A079891 A108738 A064405 * A100762 A059147 A091891

Adjacent sequences: A235869 A235870 A235871 * A235873 A235874 A235875

Keyword

nonn,mult

Author

José María Grau Ribas, Apr 03 2014

Status

approved