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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 (list; graph; refs; listen; history; text; internal format)
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
Sequence in context: A237748 A359863 A326233 * A174197 A071224 A305381
KEYWORD
nonn,mult
AUTHOR
Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003
EXTENSIONS
More terms from David Wasserman, Jun 17 2005
STATUS
approved

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Last modified June 15 06:51 EDT 2024. Contains 373402 sequences. (Running on oeis4.)