A050458 Difference between Sum_{d|n, d == 1 (mod 4)} d^2 and Sum_{d|n, d == 3 (mod 4)} d^2.

1, 1, 8, 1, 26, 8, 48, 1, 73, 26, 120, 8, 170, 48, 208, 1, 290, 73, 360, 26, 384, 120, 528, 8, 651, 170, 656, 48, 842, 208, 960, 1, 960, 290, 1248, 73, 1370, 360, 1360, 26, 1682, 384, 1848, 120, 1898, 528, 2208, 8, 2353, 651, 2320, 170, 2810, 656, 3120, 48, 2880, 842

Offset: 1

Author

Vladeta Jovovic, Feb 15 2004

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

a(n) = |A002173(n)|.

Programs

Mathematica

f[p_, e_] := If[Mod[p, 4] == 1, ((p^2)^(e+1)-1)/(p^2-1), ((p^2)^(e+1)+(-1)^e)/(p^2+1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Aug 28 2023 *)

PARI

{a(n)=if(n<1, 0, abs(sumdiv(n, d, d^2*kronecker(-4,d))))} /* Michael Somos, Aug 09 2006 */

PARI

{a(n)= local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 1, if(p%4==1, ((p^2)^(e+1)-1)/(p^2-1), ((p^2)^(e+1)+(-1)^e)/(p^2+1)))))) } /* Michael Somos, Aug 09 2006 */

PARI

{a(n)=if(n<1, 0, polcoeff( sum(k=1,n, x^k/(1+x^(2*k))*(k/2^valuation(k,2))^2, x*O(x^n)), n))} /* Michael Somos, Aug 09 2006 */

Formula

Multiplicative with a(p^e) = 1 if p = 2; ((p^2)^(e+1)-1)/(p^2-1) if p == 1 (mod 4); ((p^2)^(e+1)+(-1)^e)/(p^2+1) if p == 3 (mod 4). - Michael Somos, Aug 09 2006

Extensions

Edited by Michael Somos, Aug 09 2006