OFFSET
1,2
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
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
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, Nov 25 2006
STATUS
approved