A087561 Number of solutions to x^2 + 2y^2 == 0 (mod n).

1, 2, 5, 4, 1, 10, 1, 8, 21, 2, 21, 20, 1, 2, 5, 16, 33, 42, 37, 4, 5, 42, 1, 40, 25, 2, 81, 4, 1, 10, 1, 32, 105, 66, 1, 84, 1, 74, 5, 8, 81, 10, 85, 84, 21, 2, 1, 80, 49, 50, 165, 4, 1, 162, 21, 8, 185, 2, 117, 20, 1, 2, 21, 64, 1, 210, 133, 132, 5, 2, 1, 168, 145, 2, 125, 148, 21

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 24 2003

Keywords

• mult
• nonn
• easy

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..10000
• László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), Article 14.11.6.

Crossrefs

Cf. A086933, A088964, A088965, A089002, A309710.

Programs

Mathematica

a[n_] := If[n == 1, 1, Product[{p, e} = pe; Which[p == 2, 2^e, Abs[Mod[p, 8] - 2] != 1, (p^2)^Quotient[e, 2], True, (p + e (p-1)) p^(e-1)], {pe, FactorInteger[n]}]];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019, from PARI *)

PARI

a(n)={my(v=vector(n)); for(i=0, n-1, v[i^2%n + 1]++); sum(i=0, n-1, v[i+1]*v[(-2*i)%n + 1])} \\ Andrew Howroyd, Jul 16 2018

PARI

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 2^e, if(abs(p%8-2)<>1, (p^2)^(e\2), (p+e*(p-1))*p^(e-1))))} \\ Andrew Howroyd, Jul 16 2018

Formula

Multiplicative with a(2^e) = 2^e, a(p^e) = p^(2*floor(e/2)) for p - 2 == +-3 (mod 8), a(p^e) = ((p-1)*e+p)*p^(e-1) for p - 2 == +-1 (mod 8). - Andrew Howroyd, Jul 16 2018

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi/(4*sqrt(2)*A309710) = 0.521595326207... . - Amiram Eldar, Nov 21 2023

Extensions

More terms from David Wasserman, Jun 07 2005