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

Author

N. J. A. Sloane

Keywords

• nonn
• mult
• easy

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).

Crossrefs

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

Programs

Magma

[DivisorSigma(6,n): n in [1..30]]; // Bruno Berselli, Apr 10 2013

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 *)
DivisorSigma[6,Range[30]] (* Harvey P. Dale, May 11 2025 *)

PARI

a(n)=sigma(n,6) \\ Charles R Greathouse IV, Apr 28 2011

Sage

[sigma(n,6)for n in range(1,24)] # Zerinvary Lajos, Jun 04 2009

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)