A000499 a(n) = Sum_{k=1..n-1} k^3*sigma(k)*sigma(n-k).

0, 1, 27, 184, 875, 2700, 7546, 17600, 35721, 72750, 126445, 223776, 353717, 595448, 843750, 1349120, 1827636, 2808837, 3600975, 5306000, 6667920, 9599172, 11509982, 16416000, 19015625, 26605670, 30902310, 41686848, 46948825, 64233000, 70306760, 94089216

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			G.f. = x^2 + 27*x^3 + 184*x^4 + 875*x^5 + 2700*x^6 + 7546*x^7 + 17600*x^8 + ...
		

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

• John Cerkan, Table of n, a(n) for n = 1..10000
• Jacques Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]

Crossrefs

Cf. A000385, A000441, A000477, A259692, A259693, A259694, A259695, A259696.

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

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(3);

Mathematica

a[n_] := Sum[k^3*DivisorSigma[1, k]*DivisorSigma[1, n - k], {k, 1, n - 1}]; Array[a, 32] (* Jean-François Alcover, Feb 09 2016 *)

PARI

a(n) = sum(k=1, n-1, k^3*sigma(k)*sigma(n-k)); \\ Michel Marcus, Feb 02 2014

PARI

a(n) = my(f = factor(n)); ((n^3 - 3*n^4) * sigma(f) + 2*n^3 * sigma(f, 3)) / 24; \\ Amiram Eldar, Jan 04 2025

Formula

a(n) = Sum_{k=1..n-1} k^3*sigma(k)*sigma(n-k). - Michel Marcus, Feb 02 2014

a(n) = (n^3/24 - n^4/8)*sigma_1(n) + (n^3/12)*sigma_3(n). - Ridouane Oudra, Sep 15 2020

Sum_{k=1..n} a(k) ~ Pi^4 * n^7 / 7560. - Vaclav Kotesovec, Aug 08 2022

Extensions

More terms and 0 prepended by Michel Marcus, Feb 02 2014