%I #40 Oct 29 2023 02:34:34
%S 1,65,730,4161,15626,47450,117650,266305,532171,1015690,1771562,
%T 3037530,4826810,7647250,11406980,17043521,24137570,34591115,47045882,
%U 65019786,85884500,115151530,148035890,194402650,244156251,313742650,387952660,489541650,594823322,741453700
%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
%F From _Amiram Eldar_, Oct 29 2023: (Start)
%F Multiplicative with a(p^e) = (p^(6*e+6)-1)/(p^6-1).
%F Dirichlet g.f.: zeta(s)*zeta(s-6).
%F Sum_{k=1..n} a(k) = zeta(7) * n^7 / 7 + O(n^8). (End)
%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
%Y Cf. A000203, A001014, A001157-A001160, A013665, A013955-A013972, A017665-A017712.
%K nonn,mult,easy
%O 1,2
%A _N. J. A. Sloane_