A000499 a(n) = Sum_{k=1..n-1} k^3*sigma(k)*sigma(n-k). (Formerly M5193 N2257)

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

1,3

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

John Cerkan, Table of n, a(n) for n = 1..10000

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

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

Example

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

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 *)

Prog

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

Crossrefs

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

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

Sequence in context: A224454 A258637 A228463 * A365890 A365884 A042416

Adjacent sequences: A000496 A000497 A000498 * A000500 A000501 A000502

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved