A082245 Sum of (n-1)-th powers of divisors of n.

1, 3, 10, 73, 626, 8052, 117650, 2113665, 43053283, 1001953638, 25937424602, 743375541244, 23298085122482, 793811662272744, 29192932133689220, 1152956690052710401, 48661191875666868482, 2185928253847184914509

Offset: 1

Author

Reinhard Zumkeller, May 22 2003

Keywords

• nonn

Comments

a(n) = t(n,n-1), t as defined in A082771;

a(1)=A000005(1), a(2)=A000203(2), a(3)=A001157(3), a(4)=A001158(4), a(5)=A001159(5), a(6)=A001160(6), a(7)=A013954(7), a(8)=A013955(8).

Examples

			a(6) = 1^5 + 2^5 + 3^5 + 6^5 = 1 + 32 + 243 + 7776 = 8052.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..387
• M. Sugunamma, Certain results concerning sigma_k(n) and phi_k(n), Annales Polonici Mathematici, Vol. 8, No. 2 (1960), pp. 173-176.
• Eric Weisstein's World of Mathematics, Divisor Function.

Crossrefs

Cf. A023887, A000005, A000203, A001157, A001158, A001159, A001160, A013954, A013955, A013956, A013957, A013958

Cf. A013959, A013960, A013961, A013962, A013963, A013964, A013965, A013966, A013967, A013968, A013969, A013970

Cf. A001113, A013971, A013972.

Programs

Magma

[DivisorSigma(n-1, n): n in [1..20]]; // G. C. Greubel, Nov 02 2018

Mathematica

Table[Total[Divisors[n]^(n-1)], {n,18}] (* T. D. Noe, Oct 25 2006 *)
Table[DivisorSigma[n-1,n], {n,1,20}] (* G. C. Greubel, Nov 02 2018 *)

PARI

a(n) = sigma(n, n-1); \\ Michel Marcus, Nov 07 2017

PARI

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-(k*x)^k)^(1/k^2))))) \\ Seiichi Manyama, Jun 23 2019

Sage

[sigma(n,(n-1))for n in range(1,19)] # Zerinvary Lajos, Jun 04 2009

Formula

G.f.: Sum_{k>=1} k^(k-1)*x^k/(1 - (k*x)^k). - Ilya Gutkovskiy, Nov 02 2018

L.g.f.: -log(Product_{k>=1} (1 - (k*x)^k)^(1/k^2)) = Sum_{k>=1} a(k)*x^k/k. - Seiichi Manyama, Jun 23 2019

Limit_{n->oo} a(n)/A023887(n-1) = e (A001113) (Sugunamma, 1960). - Amiram Eldar, Apr 15 2021

Extensions

Corrected by T. D. Noe, Oct 25 2006