A000441 a(n) = Sum_{k=1..n-1} k*sigma(k)*sigma(n-k).

0, 1, 9, 34, 95, 210, 406, 740, 1161, 1920, 2695, 4116, 5369, 7868, 9690, 13640, 16116, 22419, 25365, 34160, 38640, 50622, 55154, 73320, 77225, 100100, 107730, 135576, 141085, 182340, 184760, 233616, 243408, 297738, 301420, 385110, 377511, 467210, 478842

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Apart from initial zero this is the convolution of A340793 and A143128. - Omar E. Pol, Feb 16 2021

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).
• Jacques 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
• Jacques Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]

Crossrefs

Cf. A000385, A000477, A000499, A259692, A259693, A259694, A259695, A259696.

Cf. A000203 (sigma_1), A001158 (sigma_3), A064987.

Cf. A143128, A340793.

Programs

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-François Alcover, Feb 08 2016 *)

PARI

a(n) = sum(k=1, n-1, k*sigma(k)*sigma(n-k)); \\ Michel Marcus, Feb 02 2014

PARI

a(n) = my(f = factor(n)); ((n - 6*n^2) * sigma(f) + 5*n * sigma(f, 3)) / 24; \\ Amiram Eldar, Jan 04 2025

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

Formula

Convolution of A000203 with A064987. - Sean A. Irvine, Nov 14 2010

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

Extensions

More terms from Sean A. Irvine, Nov 14 2010

a(1)=0 prepended by Michel Marcus, Feb 02 2014