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A013954 sigma_6(n), the sum of the 6th powers of the divisors of n. 86
1, 65, 730, 4161, 15626, 47450, 117650, 266305, 532171, 1015690, 1771562, 3037530, 4826810, 7647250, 11406980, 17043521, 24137570, 34591115, 47045882, 65019786, 85884500, 115151530, 148035890 (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. - comment from 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

FORMULA

G.f. sum(k>=1, k^6*x^k/(1-x^k)). - Benoit Cloitre, Apr 21 2003

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 [From Vladimir Joseph Stephan Orlovsky, Mar 11 2009]

PROG

(Sage) [sigma(n, 6)for n in xrange(1, 24)] # [From 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: A088677 A034680 A017675 * A116277 A220389 A196634

Adjacent sequences:  A013951 A013952 A013953 * A013955 A013956 A013957

KEYWORD

nonn,mult,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified October 23 14:38 EDT 2014. Contains 248465 sequences.