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A023887 a(n) = sigma_n(n): sum of n-th powers of divisors of n. 27
1, 5, 28, 273, 3126, 47450, 823544, 16843009, 387440173, 10009766650, 285311670612, 8918294543346, 302875106592254, 11112685048647250, 437893920912786408, 18447025552981295105, 827240261886336764178, 39346558271492178925595, 1978419655660313589123980 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Logarithmic derivative of A023881.

Compare to A217872(n) = sigma(n)^n.

REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.

LINKS

N. Hobson, Table of n, a(n) for n = 1..100

FORMULA

G.f.: Sum_{n>0} (n*x)^n/(1-(n*x)^n). - Vladeta Jovovic, Oct 27 2002

If the canonical prime factorization of n > 1 is the product of p^e(p) then sigma_n(n) = Product_p ((p^(n*(e(p)+1)))-1)/(p^n-1). - Nick Hobson, Nov 25 2006

sigma_n(n) is odd if and only if n is a square or twice a square. - Nick Hobson, Nov 25 2006

Conjecture: sigma_m(n) = sigma(n^m * rad(n)^(m-1))/sigma(rad(n)^(m-1)) for n > 0 and m > 0, where sigma = A000203 and rad = A007947. - Velin Yanev, Aug 24 2017

a(n) ~ n^n. - Vaclav Kotesovec, Nov 02 2018

EXAMPLE

The divisors of 6 are 1, 2, 3 and 6, so a(6) = 1^6 + 2^6 + 3^6 + 6^6 = 47450.

MATHEMATICA

Table[DivisorSigma[n, n], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)

PROG

(PARI) a(n) = if(n<1, 0, sigma(n, n)) \\ Nick Hobson, Nov 25 2006

(Maxima) makelist(divsum(n, n), n, 1, 20); \\ Emanuele Munarini, Mar 26 2011

CROSSREFS

Cf. A000203, A001157-A001160, A013954-A013972, A023881.

Sequence in context: A062796 A224607 A320974 * A171187 A057792 A174464

Adjacent sequences:  A023884 A023885 A023886 * A023888 A023889 A023890

KEYWORD

nonn

AUTHOR

Olivier Gérard

EXTENSIONS

Edited by N. J. A. Sloane, Nov 25 2006

STATUS

approved

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Last modified October 18 10:39 EDT 2019. Contains 328147 sequences. (Running on oeis4.)