A240547 Number of non-congruent solutions of x^2 + y^2 + z^2 + t^2 == 0 mod n.

1, 8, 33, 32, 145, 264, 385, 128, 945, 1160, 1441, 1056, 2353, 3080, 4785, 512, 5185, 7560, 7201, 4640, 12705, 11528, 12673, 4224, 18625, 18824, 26001, 12320, 25201, 38280, 30721, 2048, 47553, 41480, 55825, 30240, 51985, 57608, 77649, 18560, 70561, 101640

Offset: 1

Author

Laszlo Toth, Apr 07 2014

Keywords

• nonn
• mult

Examples

			For n=2 the a(2)=8 solutions are (0,0,0,0), (1,1,0,0), (1,0,1,0), (1,0,0,1), (0,1,1,0), (0,1,0,1), (0,0,1,1), (1,1,1,1).
		

Links

• David A. Corneth, 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, J. Int. Seq. 17 (2014), Article 14.11.6.
• Index to sequences related to sums of squares.

Crossrefs

Cf. A020478, A069097, A086933, A087687, A208895, A229179.

Programs

Maple

A240547 := proc(n) local a, x, y, z, t ; a := 0 ; for x from 0 to n-1 do for y
from 0 to n-1 do for z from 0 to n-1 do for t from 0 to n-1 do if
(x^2+y^2+z^2+t^2) mod n = 0 mod n then a := a+1 ; fi; od; od ; od; od;
a ; end proc;
# alternative
A240547 := proc(n)
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if p = 2 then
            a := a*p^(2*e+1) ;
        else
            a := a* p^(2*e-1)*(p^(e+1)+p^e-1) ;
        end if;
    end do:
    a ;
end proc:
seq(A240547(n),n=1..100) ; # R. J. Mathar, Jun 25 2018

Mathematica

b[2, e_] := 2^(2 e + 1);
b[p_, e_] := p^(2 e - 1)*(p^(e + 1) + p^e - 1);
a[n_] := Times @@ b @@@ FactorInteger[n];
Array[a, 42] (* Jean-François Alcover, Dec 05 2017 *)

PARI

a(n) = my(m); if( n<1, 0, forvec( v = vector(4, i, [0, n-1]), m += (0 == norml2(v)%n))); m /* Michael Somos, Apr 07 2014 */

PARI

a(n) = {my(f = factor(n), res = 1, start = 1, p, e, i); if(n % 2 == 0, res = 1<<(f[1,2]<<1+1); start = 2); for(i = start, #f~, p = f[i, 1]; e = f[i, 2]; res*=(p^(e<<1-1)*(p^(e+1)+p^e-1))); res} \\ David A. Corneth, Jul 22 2018

Formula

Multiplicative, with a(2^e) = 2^(2e+1) for e>=1, a(p^e) = p^(2e-1)*(p^(e+1)+p^e-1) for p > 2, e>=1.

For odd n, a(n) = A069097(n)*n = A020478(n). - R. J. Mathar, Jun 23 2018

Sum_{k=1..n} a(k) ~ c * n^4 + O(n^3 * log(n)), where c = 5*Pi^2/(168*zeta(3)) = 0.244362... (Tóth, 2014). - Amiram Eldar, Oct 18 2022