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

1,2

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.

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

Example

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).

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A079271 A336220 A247533 * A031445 A131547 A044085

Adjacent sequences: A240544 A240545 A240546 * A240548 A240549 A240550

Keyword

nonn,mult

Author

Laszlo Toth, Apr 07 2014

Status

approved