login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A087784
Number of solutions to x^2 + y^2 + z^2 = 1 mod n.
12
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
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
KEYWORD
nonn,mult
AUTHOR
Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003
EXTENSIONS
More terms from David Wasserman, Jun 17 2005
STATUS
approved