A010786 Floor-factorial numbers: a(n) = Product_{k=1..n} floor(n/k).

1, 1, 2, 3, 8, 10, 36, 42, 128, 216, 600, 660, 3456, 3744, 9408, 18900, 61440, 65280, 279936, 295488, 1152000, 2116800, 4878720, 5100480, 31850496, 41472000, 93450240, 163762560, 568995840, 589317120, 3265920000, 3374784000, 11324620800, 19269550080, 42188636160

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• nice

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]

Examples

			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]
		

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
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A006218, A075885, A092143, A131385, A131387, A377484, A007955, A075993.

Programs

Haskell

a010786 n = product $ map (div n) [1..n]
-- Reinhard Zumkeller, Feb 26 2012

Magma

[&*[n div i: i in [1..n]]: n in [1..35]]; // Vincenzo Librandi, Oct 03 2018

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 *)

PARI

vector(50, n, prod(k=1, n, n\k)) \\ Michel Marcus, Nov 10 2014

Formula

a(n+1) = a(n)*A208449(n)/A208450(n). - Reinhard Zumkeller, Feb 26 2012

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)

From Ridouane Oudra, Jan 18 2025: (Start)

a(n) = Product_{k=1..n} ((k+1)/k)^floor(n/(k+1)).

a(n) = Product_{k=1..n} k^A075993(n, k).

a(n) = A092143(n)/f(n), where f(n) = Product_{k=1..n} ((floor(n/k)-1)!).

a(n) = A092143(n)/g(n), where g(n) = Product_{k=1..n} A377484(k).

a(n)/a(n-1) = A007955(n)/A377484(n). (End)

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