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A023887
a(n) = sigma_n(n): sum of n-th powers of divisors of n.
58
1, 5, 28, 273, 3126, 47450, 823544, 16843009, 387440173, 10009766650, 285311670612, 8918294543346, 302875106592254, 11112685048647250, 437893920912786408, 18447025552981295105, 827240261886336764178, 39346558271492178925595, 1978419655660313589123980
OFFSET
1,2
COMMENTS
Logarithmic derivative of A023881.
Compare to A217872(n) = sigma(n)^n.
REFERENCES
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
LINKS
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
EXAMPLE
The divisors of 6 are 1, 2, 3 and 6, so a(6) = 1^6 + 2^6 + 3^6 + 6^6 = 47450.
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 *)
PROG
(PARI) a(n) = sigma(n, n); \\ Nick Hobson, Nov 25 2006
(Maxima) makelist(divsum(n, n), n, 1, 20); \\ Emanuele Munarini, Mar 26 2011
(Python)
from sympy import divisor_sigma
def A023887(n): return divisor_sigma(n, n) # Chai Wah Wu, Jun 19 2022
CROSSREFS
KEYWORD
nonn,easy
EXTENSIONS
Edited by N. J. A. Sloane, Nov 25 2006
STATUS
approved