A000441 a(n) = Sum_{k=1..n-1} k*sigma(k)*sigma(n-k). (Formerly M4613 N1968)

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

1,3

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]

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

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 *)

Prog

(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

Crossrefs

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

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

Cf. A143128, A340793.

Sequence in context: A326278 A014816 A147691 * A067989 A002881 A268803

Adjacent sequences: A000438 A000439 A000440 * A000442 A000443 A000444

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Nov 14 2010

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

Status

approved