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

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

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for sequences related to sigma(n).

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

Cf. A000203, A001014, A001157-A001160, A013665, A013955-A013972, A017665-A017712.

Sequence in context: A034680 A351301 A017675 * A294301 A343508 A116277

Adjacent sequences: A013951 A013952 A013953 * A013955 A013956 A013957

Keyword

nonn,mult,easy

Author

N. J. A. Sloane

Status

approved