A023887 a(n) = sigma_n(n): sum of n-th powers of divisors of n.

1, 5, 28, 273, 3126, 47450, 823544, 16843009, 387440173, 10009766650, 285311670612, 8918294543346, 302875106592254, 11112685048647250, 437893920912786408, 18447025552981295105, 827240261886336764178, 39346558271492178925595, 1978419655660313589123980

Offset: 1

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

Logarithmic derivative of A023881.

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

Examples

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

References

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

Links

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

Crossrefs

Cf. A000203, A001157-A001160, A013954-A013972, A023881, A199858, subdiagonal in A109974.

Programs

Maple

A023887 := proc(n)
    numtheory[sigma][n](n) ;
end proc:
seq(A023887(n),n=1..10) ; # R. J. Mathar, Apr 06 2022

Mathematica

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

Maxima

makelist(divsum(n,n),n,1,20); /* Emanuele Munarini, Mar 26 2011 */

PARI

a(n) = sigma(n,n); \\ Nick Hobson, Nov 25 2006

Python

from sympy import divisor_sigma
def A023887(n): return divisor_sigma(n,n) # Chai Wah Wu, Jun 19 2022

Formula

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

From Nick Hobson, Nov 25 2006: (Start)

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).

sigma_n(n) is odd if and only if n is a square or twice a square. (End)

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

Sum_{n>=1} 1/a(n) = A199858. - Amiram Eldar, Nov 19 2020

Extensions

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