A087784 Number of solutions to x^2 + y^2 + z^2 = 1 mod n.

1, 4, 6, 24, 30, 24, 42, 96, 54, 120, 110, 144, 182, 168, 180, 384, 306, 216, 342, 720, 252, 440, 506, 576, 750, 728, 486, 1008, 870, 720, 930, 1536, 660, 1224, 1260, 1296, 1406, 1368, 1092, 2880, 1722, 1008, 1806, 2640, 1620, 2024, 2162, 2304, 2058, 3000

Offset

1,2

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

László Tóth, Counting solutions of quadratic congruences in several variables revisited, arXiv preprint arXiv:1404.4214 [math.NT], 2014.

László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, Journal of Integer Sequences, 17 (2014), Article 14.11.6.

Formula

a(n) = n^2 * (3/2 if 4|n) * Product_{primes p == 1 mod 4 dividing n} (1+1/p) * Product_{primes p == 3 mod 4 dividing n} (1-1/p). - Bjorn Poonen, Dec 09 2003

Sum_{k=1..n} a(k) ~ c * n^3 + O(n^2 * log(n)), where c = 36*G/Pi^4 = 0.338518..., where G is Catalan's constant (A006752) (Tóth, 2014). - Amiram Eldar, Oct 18 2022

Mathematica

Table[With[{f = FactorInteger[n][[All, 1]]}, Apply[Times, Map[1 + 1/# &, Select[f, Mod[#, 4] == 1 &]]] Apply[Times, Map[1 - 1/# &, Select[f, Mod[#, 4] == 3 &]]] (1 + Boole[Divisible[n, 4]]/2) n^2] - Boole[n == 1], {n, 50}] (* Michael De Vlieger, Feb 15 2018 *)

Prog

(PARI) a(n) = {my(f=factor(n)); if ((n % 4), 1, 3/2)*n^2*prod(k=1, #f~, p = f[k, 1]; m = p % 4; if (m==1, 1+1/p, if (m==3, 1-1/p, 1))); } \\ Michel Marcus, Feb 14 2018

Crossrefs

Cf. A006752, A060968, A087687, A208895, A264083.

Sequence in context: A237748 A359863 A326233 * A174197 A071224 A305381

Adjacent sequences: A087781 A087782 A087783 * A087785 A087786 A087787

Keyword

nonn,mult

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003

Extensions

More terms from David Wasserman, Jun 17 2005

Status

approved