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

1,2

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).

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.

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

Example

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

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 *)

Prog

(Sage) [sigma(n, (n-1))for n in range(1, 19)] # Zerinvary Lajos, Jun 04 2009
(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
(Magma) [DivisorSigma(n-1, n): n in [1..20]]; // G. C. Greubel, Nov 02 2018

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.

Sequence in context: A337949 A086846 A277614 * A158033 A027159 A236696

Adjacent sequences: A082242 A082243 A082244 * A082246 A082247 A082248

Keyword

nonn

Author

Reinhard Zumkeller, May 22 2003

Extensions

Corrected by T. D. Noe, Oct 25 2006

Status

approved