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A001160 sigma_5(n), the sum of the 5th powers of the divisors of n.
(Formerly M5240 N2279)
107
1, 33, 244, 1057, 3126, 8052, 16808, 33825, 59293, 103158, 161052, 257908, 371294, 554664, 762744, 1082401, 1419858, 1956669, 2476100, 3304182, 4101152, 5314716, 6436344, 8253300, 9768751, 12252702, 14408200, 17766056, 20511150 (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.

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zagier, Don. "Elliptic modular forms and their applications." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103. See p. 17, G_6(z).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.

FORMULA

Multiplicative with a(p^e) = (p^(5e+5)-1)/(p^5-1). - David W. Wilson, Aug 01, 2001.

G.f. sum(k>=1, k^5*x^k/(1-x^k)). - Benoit Cloitre, Apr 21 2003

Dirichlet g.f. zeta(s)*zeta(s-5). - R. J. Mathar, Mar 06 2011

MATHEMATICA

lst={}; Do[AppendTo[lst, DivisorSigma[5, n]], {n, 5!}]; lst [From Vladimir Joseph Stephan Orlovsky, Mar 11 2009]

DivisorSigma[5, Range[30]] (* Harvey P. Dale, Nov 11 2013 *)

PROG

(Sage) [sigma(n, 5)for n in xrange(1, 30)] # [From Zerinvary Lajos, Jun 04 2009]

(PARI) a(n)=sigma(n, 5) \\ Charles R Greathouse IV, Apr 28, 2011

(MAGMA) [DivisorSigma(5, n): n in [1..30]]; // Bruno Berselli, Apr 10 2013

CROSSREFS

Cf. A000005, A000203, A001157-A001159.

Sequence in context: A088703 A034679 A017673 * A184059 A197347 A197398

Adjacent sequences:  A001157 A001158 A001159 * A001161 A001162 A001163

KEYWORD

nonn,easy,mult

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified November 29 03:09 EST 2014. Contains 250479 sequences.