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

%I A082245 #35 Feb 16 2025 08:32:48
%S A082245 1,3,10,73,626,8052,117650,2113665,43053283,1001953638,25937424602,
%T A082245 743375541244,23298085122482,793811662272744,29192932133689220,
%U A082245 1152956690052710401,48661191875666868482,2185928253847184914509
%N A082245 Sum of (n-1)-th powers of divisors of n.
%C A082245 a(n) = t(n,n-1), t as defined in A082771;
%C A082245 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).
%H A082245 Seiichi Manyama, <a href="/A082245/b082245.txt">Table of n, a(n) for n = 1..387</a>
%H A082245 M. Sugunamma, <a href="http://doi.org/10.4064/ap-8-2-173-176">Certain results concerning sigma_k(n) and phi_k(n)</a>, Annales Polonici Mathematici, Vol. 8, No. 2 (1960), pp. 173-176.
%H A082245 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>.
%F A082245 G.f.: Sum_{k>=1} k^(k-1)*x^k/(1 - (k*x)^k). - _Ilya Gutkovskiy_, Nov 02 2018
%F A082245 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
%F A082245 Limit_{n->oo} a(n)/A023887(n-1) = e (A001113) (Sugunamma, 1960). - _Amiram Eldar_, Apr 15 2021
%e A082245 a(6) = 1^5 + 2^5 + 3^5 + 6^5 = 1 + 32 + 243 + 7776 = 8052.
%t A082245 Table[Total[Divisors[n]^(n-1)], {n,18}] (* _T. D. Noe_, Oct 25 2006 *)
%t A082245 Table[DivisorSigma[n-1,n], {n,1,20}] (* _G. C. Greubel_, Nov 02 2018 *)
%o A082245 (Sage) [sigma(n,(n-1))for n in range(1,19)] # _Zerinvary Lajos_, Jun 04 2009
%o A082245 (PARI) a(n) = sigma(n, n-1); \\ _Michel Marcus_, Nov 07 2017
%o A082245 (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
%o A082245 (Magma) [DivisorSigma(n-1, n): n in [1..20]]; // _G. C. Greubel_, Nov 02 2018
%Y A082245 Cf. A023887, A000005, A000203, A001157, A001158, A001159, A001160, A013954, A013955, A013956, A013957, A013958
%Y A082245 Cf. A013959, A013960, A013961, A013962, A013963, A013964, A013965, A013966, A013967, A013968, A013969, A013970
%Y A082245 Cf. A001113, A013971, A013972.
%K A082245 nonn
%O A082245 1,2
%A A082245 _Reinhard Zumkeller_, May 22 2003
%E A082245 Corrected by _T. D. Noe_, Oct 25 2006