A208895 Number of non-congruent solutions to x^2 + y^2 + z^2 + t^2 == 1 (mod n).

1, 8, 24, 64, 120, 192, 336, 512, 648, 960, 1320, 1536, 2184, 2688, 2880, 4096, 4896, 5184, 6840, 7680, 8064, 10560, 12144, 12288, 15000, 17472, 17496, 21504, 24360, 23040, 29760, 32768, 31680, 39168, 40320, 41472, 50616, 54720, 52416, 61440, 68880, 64512

Offset

1,2

Links

Amiram Eldar, 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 Toth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), Article 14.11.6.

Formula

Conjecture: a(n) = n*Sum_{d|2*n} d^2*mu(2*n/d)/3. - Gionata Neri, Feb 18 2018

From Amiram Eldar, Oct 18 2022: (Start)

Multiplicative with a(p^e) = p^(3*e)*(1-1/p^2) if p > 2, and a(2^e) = 8^e.

Sum_{k=1..n} a(k) ~ c * n^4 + O(n^3), where c = 2/(7*zeta(3)) = 0.237687... (Tóth, 2014). (End)

Maple

A208895 := proc(n)
    local a, pe, p, nu ;
    a := 1 ;
    for pe in ifactors(n)[2] do
        p := op(1, pe) ;
        nu := op(2, pe) ;
        if p > 2 then
            a := a*p^(3*nu)*(1-1/p^2) ;
        else
            a := a*8^nu ;
        end if;
    end do:
    a ;
end proc:
seq(A208895(n), n=1..20) ; # R. J. Mathar, Jun 23 2018

Mathematica

a[n_] := Length[Union[Flatten[Table[If[Mod[x^2 + y^2 + z^2 + t^2, n] == 1, {x, y, z, t}], {x, n}, {y, n}, {z, n}, {t, n}], 3]]] - 1; Join[{1}, Table[a[n], {n, 2, 30}]]
f[p_, e_] := p^(3*e) * (1-1/p^2); f[2, e_] := 8^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Oct 18 2022 *)

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, 8^f[i, 2], f[i, 1]^(3*f[i, 2]) * (1 - 1/f[i, 1]^2))); } \\ Amiram Eldar, Oct 18 2022

Crossrefs

Cf. A060968, A087784.

Sequence in context: A066605 A066497 A205963 * A111071 A090336 A364245

Adjacent sequences: A208892 A208893 A208894 * A208896 A208897 A208898

Keyword

nonn,mult

Author

José María Grau Ribas, Mar 03 2012

Status

approved