A316148 Number of non-congruent solutions of x^2+y^2 == z^2+w^2 (mod n).

1, 8, 33, 96, 145, 264, 385, 896, 945, 1160, 1441, 3168, 2353, 3080, 4785, 7680, 5185, 7560, 7201, 13920, 12705, 11528, 12673, 29568, 18625, 18824, 26001, 36960, 25201, 38280, 30721, 63488, 47553, 41480, 55825, 90720, 51985, 57608, 77649, 129920, 70561, 101640, 81313

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Index to sequences related to sums of squares.

Formula

Multiplicative with a(2^e) = 2^(2e+1)*(2^e-1), a(p^e) = p^(3e)+p^(3e-1)-p^(2e-1) for odd primes p.

Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(2)/zeta(3) = 1.368432... (A306633). - Amiram Eldar, Dec 18 2023

Maple

A316148 := 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)*(p^e-1) ;
        else
            a := a*p^(2*e-1)*(p^(e+1)+p^e-1) ;
        end if;
    end do:
    a ;
end proc:
seq(A316148(n), n=1..100) ;

Mathematica

f[2, e_] :=  2^(2*e+1)*(2^e-1); f[p_, e_] := p^(3*e)+p^(3*e-1)-p^(2*e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)

Prog

(PARI) a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(p == 2, 2^(2*e+1)*(2^e-1), p^(3*e)+p^(3*e-1)-p^(2*e-1))); } \\ Amiram Eldar, Dec 18 2023

Crossrefs

Cf. A096018, A240547, A306633.

Sequence in context: A212574 A210698 A114105 * A014820 A070736 A051836

Adjacent sequences: A316145 A316146 A316147 * A316149 A316150 A316151

Keyword

nonn,easy,mult

Author

R. J. Mathar, Jun 25 2018

Status

approved