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A082245 Sum of (n-1)-th powers of divisors of n. 8
1, 3, 10, 73, 626, 8052, 117650, 2113665, 43053283, 1001953638, 25937424602, 743375541244, 23298085122482, 793811662272744, 29192932133689220, 1152956690052710401, 48661191875666868482, 2185928253847184914509 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A337949 A086846 A277614 * A158033 A027159 A236696
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, May 22 2003
EXTENSIONS
Corrected by T. D. Noe, Oct 25 2006
STATUS
approved

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Last modified February 29 14:32 EST 2024. Contains 370425 sequences. (Running on oeis4.)