A174937 a(n) = Sum_{d|n} d^tau(d).

1, 5, 10, 69, 26, 1310, 50, 4165, 739, 10030, 122, 2987358, 170, 38470, 50660, 1052741, 290, 34014263, 362, 64010094, 194540, 234382, 530, 110078305630, 15651, 457150, 532180, 481928838, 842, 656100061960

Offset: 1

Author

Jaroslav Krizek and Paul D. Hanna, Apr 02 2010

Keywords

• nonn

Comments

Here tau(n) = A000005(n) = the number of divisors of n.

Examples

			For n = 4, A007955(n) = b(n): a(4) = b(1)^2 + b(2)^2 + b(4)^2 = 1^2 + 2^2 + 8^2 = 69.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005 (tau), A174472, A174473, A345270, A345271.

Programs

Mathematica

a[n_] := DivisorSum[n, #^DivisorSigma[0, #] &]; Array[a, 30] (* Amiram Eldar, Oct 08 2021 *)

PARI

{a(n)=sumdiv(n,d,d^sigma(d,0))}

PARI

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k^numdiv(k)*x^k/(1-x^k))) \\ Seiichi Manyama, Oct 14 2021

Formula

Also a(n) = Sum_{d|n} A007955(d)^2, where A007955(m) = product of divisors of m.

Logarithmic derivative of A174473.

G.f.: Sum_{k>=1} k^tau(k) * x^k/(1 - x^k). - Seiichi Manyama, Oct 14 2021