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A013954 a(n) = sigma_6(n), the sum of the 6th powers of the divisors of n. 92

%I

%S 1,65,730,4161,15626,47450,117650,266305,532171,1015690,1771562,

%T 3037530,4826810,7647250,11406980,17043521,24137570,34591115,47045882,

%U 65019786,85884500,115151530,148035890

%N a(n) = sigma_6(n), the sum of the 6th powers of the divisors of n.

%C If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

%C Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

%C Inverse Mobius transform of A001014. - _R. J. Mathar_, Oct 13 2011

%H Vincenzo Librandi, <a href="/A013954/b013954.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F G.f.: sum_{k>=1} k^6*x^k/(1-x^k). - _Benoit Cloitre_, Apr 21 2003

%F L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^5)) = Sum_{n>=1} a(n)*x^n/n. - _Ilya Gutkovskiy_, May 06 2017

%p A013954 := proc(n)

%p numtheory[sigma][6](n) ;

%p end proc: # _R. J. Mathar_, Oct 13 2011

%t lst={};Do[AppendTo[lst,DivisorSigma[6,n]],{n,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Mar 11 2009 *)

%o (Sage) [sigma(n,6)for n in range(1,24)] # _Zerinvary Lajos_, Jun 04 2009

%o (PARI) a(n)=sigma(n,6) \\ _Charles R Greathouse IV_, Apr 28, 2011

%o (MAGMA) [DivisorSigma(6,n): n in [1..30]]; // _Bruno Berselli_, Apr 10 2013

%K nonn,mult,easy

%O 1,2

%A _N. J. A. Sloane_

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Last modified March 2 06:30 EST 2021. Contains 341743 sequences. (Running on oeis4.)