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

Author

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

Keywords

• nonn
• mult

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.

Crossrefs

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

Programs

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 *)

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

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

Extensions

More terms from David Wasserman, Jun 17 2005