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
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms and 0 prepended by Michel Marcus, Feb 02 2014
STATUS
approved