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

Author

R. J. Mathar, Jun 25 2018

Keywords

• nonn
• easy
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index to sequences related to sums of squares.

Crossrefs

Cf. A096018, A240547, A306633.

Programs

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 *)

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

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