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

0,3

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

Index entries for sequences related to factorial numbers

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)

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

Cf. A006218, A075885, A092143, A131385, A131387.

Sequence in context: A165153 A121989 A320843 * A248822 A005727 A354756

Adjacent sequences: A010783 A010784 A010785 * A010787 A010788 A010789

Keyword

nonn,nice

Author

N. J. A. Sloane, Simon Plouffe

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