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A240547 Number of non-congruent solutions of x^2 + y^2 + z^2 + t^2 == 0 mod n. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

David A. Corneth, Table of n, a(n) for n = 1..10000

L. Tóth, Counting solutions of quadratic congruences in several variables revisited, arXiv preprint arXiv:1404.4214 [math.NT], 2014.

L. Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014) # 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

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. 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

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Last modified April 20 15:38 EDT 2021. Contains 343135 sequences. (Running on oeis4.)