OFFSET
1,3
COMMENTS
REFERENCES
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).
J. Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
J. Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]
FORMULA
G.f.: x*f(x)*f'(x), where f(x) = Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 28 2018
a(n) = (n/24 - n^2/4)*sigma_1(n) + (5*n/24)*sigma_3(n). - Ridouane Oudra, Sep 17 2020
Sum_{k=1..n} a(k) ~ Pi^4 * n^5 / 2160. - Vaclav Kotesovec, May 09 2022
MAPLE
S:=(n, e)->add(k^e*sigma(k)*sigma(n-k), k=1..n-1);
f:=e->[seq(S(n, e), n=1..30)]; f(1); # N. J. A. Sloane, Jul 03 2015
MATHEMATICA
a[n_] := Sum[k*DivisorSigma[1, k]*DivisorSigma[1, n-k], {k, 1, n-1}]; Array[a, 40] (* Jean-Fran�ois Alcover, Feb 08 2016 *)
PROG
(PARI) a(n) = sum(k=1, n-1, k*sigma(k)*sigma(n-k)); \\ Michel Marcus, Feb 02 2014
(Python)
from sympy import divisor_sigma
def A000441(n): return (n*(1-6*n)*divisor_sigma(n)+5*n*divisor_sigma(n, 3))//24 # Chai Wah Wu, Jul 25 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Sean A. Irvine, Nov 14 2010
a(1)=0 prepended by Michel Marcus, Feb 02 2014
STATUS
approved