A174468 a(n) = Sum_{d|n} d*sigma(n/d)*sigma(d).

1, 9, 16, 53, 36, 144, 64, 261, 178, 324, 144, 848, 196, 576, 576, 1173, 324, 1602, 400, 1908, 1024, 1296, 576, 4176, 986, 1764, 1744, 3392, 900, 5184, 1024, 5013, 2304, 2916, 2304, 9434, 1444, 3600, 3136, 9396, 1764, 9216, 1936, 7632, 6408, 5184, 2304

Offset: 1

Author

Paul D. Hanna, Apr 04 2010

Keywords

• nonn
• easy
• mult

Comments

Logarithmic derivative of A174467.

Dirichlet convolution of A064987 and A000203. - R. J. Mathar, Feb 06 2011

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A174467, A000203 (sigma), A064987.

Programs

Mathematica

Table[Sum[d*DivisorSigma[1,d]*DivisorSigma[1,n/d], {d, Divisors[n]}], {n, 1, 50}] (* Vaclav Kotesovec, Feb 02 2019 *)
f[p_, e_] := (p^(2*e + 4) - (e + 2)*(p^2 - 1)*p^(e + 1) - 1)/((p - 1)^3*(p + 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)

PARI

{a(n)=sumdiv(n,d,d*sigma(n/d)*sigma(d))}

PARI

a(n)=sumdiv(n, x, x * sumdiv(x, y, sumdiv(y, z, z ) ) ); /* Joerg Arndt, Oct 07 2012 */

Formula

a(n) = sigma(n)^2 iff n is squarefree.

Dirichlet g.f. zeta(s) * (zeta(s-1))^2 * zeta(s-2). - R. J. Mathar, Feb 06 2011

Sum_{k=1..n} a(k) ~ Pi^4 * zeta(3) * n^3 / 108. - Vaclav Kotesovec, Feb 02 2019

Multiplicative with a(p^e) = (p^(2*e+4) - (e+2)*(p^2-1)*p^(e+1) - 1)/((p-1)^3*(p+1)). - Amiram Eldar, Aug 27 2023