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A235872
Number of solutions to the equation x^2=0 in the ring of Gaussian integers modulo n.
1
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
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
Sequence in context: A079891 A108738 A064405 * A100762 A059147 A091891
KEYWORD
nonn,mult
AUTHOR
STATUS
approved