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A013954 a(n) = sigma_6(n), the sum of the 6th powers of the divisors of n. 93
1, 65, 730, 4161, 15626, 47450, 117650, 266305, 532171, 1015690, 1771562, 3037530, 4826810, 7647250, 11406980, 17043521, 24137570, 34591115, 47045882, 65019786, 85884500, 115151530, 148035890, 194402650, 244156251, 313742650, 387952660, 489541650, 594823322, 741453700 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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).
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
Inverse Mobius transform of A001014. - R. J. Mathar, Oct 13 2011
LINKS
FORMULA
G.f.: Sum_{k>=1} k^6*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
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
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(6*e+6)-1)/(p^6-1).
Dirichlet g.f.: zeta(s)*zeta(s-6).
Sum_{k=1..n} a(k) = zeta(7) * n^7 / 7 + O(n^8). (End)
MAPLE
A013954 := proc(n)
numtheory[sigma][6](n) ;
end proc: # R. J. Mathar, Oct 13 2011
MATHEMATICA
lst={}; Do[AppendTo[lst, DivisorSigma[6, n]], {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
PROG
(Sage) [sigma(n, 6)for n in range(1, 24)] # Zerinvary Lajos, Jun 04 2009
(PARI) a(n)=sigma(n, 6) \\ Charles R Greathouse IV, Apr 28, 2011
(Magma) [DivisorSigma(6, n): n in [1..30]]; // Bruno Berselli, Apr 10 2013
CROSSREFS
Sequence in context: A034680 A351301 A017675 * A294301 A343508 A116277
KEYWORD
nonn,mult,easy
AUTHOR
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)