A174937 - OEIS

%I #16 Oct 14 2021 08:43:21

%S 1,5,10,69,26,1310,50,4165,739,10030,122,2987358,170,38470,50660,

%T 1052741,290,34014263,362,64010094,194540,234382,530,110078305630,

%U 15651,457150,532180,481928838,842,656100061960

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

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

%H Seiichi Manyama, <a href="/A174937/b174937.txt">Table of n, a(n) for n = 1..10000</a>

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

%F Logarithmic derivative of A174473.

%F G.f.: Sum_{k>=1} k^tau(k) * x^k/(1 - x^k). - _Seiichi Manyama_, Oct 14 2021

%e 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.

%t a[n_] := DivisorSum[n, #^DivisorSigma[0, #] &]; Array[a, 30] (* _Amiram Eldar_, Oct 08 2021 *)

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

%o (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

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

%K nonn

%O 1,2

%A _Jaroslav Krizek_ and _Paul D. Hanna_, Apr 02 2010