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

1,2

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.

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A116516 A011417 A231890 * A309771 A009738 A268647

Adjacent sequences: A087558 A087559 A087560 * A087562 A087563 A087564

Keyword

mult,nonn,easy

Author

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

Extensions

More terms from David Wasserman, Jun 07 2005

Status

approved