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A001160
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sigma_5(n), the sum of the 5th powers of the divisors of n.
(Formerly M5240 N2279)
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101
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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
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OFFSET
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1,2
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COMMENTS
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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.
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REFERENCES
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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).
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LINKS
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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.
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FORMULA
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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
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MATHEMATICA
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lst={}; Do[AppendTo[lst, DivisorSigma[5, n]], {n, 5!}]; lst [From Vladimir Joseph Stephan Orlovsky, Mar 11 2009]
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PROG
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(Sage) [sigma(n, 5)for n in xrange(1, 30)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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
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CROSSREFS
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Cf. A000005, A000203, A001157-A001159.
Sequence in context: A088703 A034679 A017673 * A184059 A197347 A197398
Adjacent sequences: A001157 A001158 A001159 * A001161 A001162 A001163
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KEYWORD
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nonn,easy,mult
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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