OFFSET
0,3
COMMENTS
Product floor(n/1)*floor(n/2)*floor(n/3)*...*floor(n/n).
a(n) is the number of functions f:[n]->[n] where f(x) is a multiple of x for all x in [n]. We note that there are floor[n/x] possible choices for each image of x under f. [Dennis P. Walsh, Nov 06 2014]
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)
Eric Weisstein's World of Mathematics, Alladi-Grinstead Constant
FORMULA
GCD(a(n), a(n+1)) = A208448(n). - Reinhard Zumkeller, Feb 26 2012
From Vaclav Kotesovec, Oct 03 2018: (Start)
log(a(n)) ~ c * (n - log(2*Pi*n)/2), where c = 0.7885...
Conjecture: c = A085361. (End)
EXAMPLE
For n=4 the a(4)=8 functions are given by the image sequences <1,2,3,4>, <1,4,3,4>, <2,2,3,4>, <2,4,3,4>, <3,2,3,4>, <3,4,3,4>, <4,2,3,4>, and <4,4,3,4>. [Dennis P. Walsh, Nov 06 2014]
MAPLE
a := n -> mul( floor(n/k), k=1..n);
MATHEMATICA
Table[Product[Floor[n/k], {k, n}], {n, 40}] (* Harvey P. Dale, May 09 2017 *)
PROG
(Haskell)
a010786 n = product $ map (div n) [1..n]
-- Reinhard Zumkeller, Feb 26 2012
(PARI) vector(50, n, prod(k=1, n, n\k)) \\ Michel Marcus, Nov 10 2014
(Magma) [&*[n div i: i in [1..n]]: n in [1..35]]; // Vincenzo Librandi, Oct 03 2018
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Hieronymus Fischer, Jul 08 2007
Edited by N. J. A. Sloane, Jul 05 2008 at the suggestion of Rick L. Shepherd
a(0)=1 prepended by Alois P. Heinz, Oct 30 2023
STATUS
approved