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

Author

José María Grau Ribas, Mar 03 2012

Keywords

• nonn
• mult

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.

Crossrefs

Cf. A060968, A087784.

Programs

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 *)

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

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)