A131385 Product ceiling(n/1)*ceiling(n/2)*ceiling(n/3)*...*ceiling(n/n) (the 'ceiling factorial').

1, 1, 2, 6, 16, 60, 144, 672, 1536, 6480, 19200, 76032, 165888, 1048320, 2257920, 8294400, 28311552, 126904320, 268738560, 1470873600, 3096576000, 16094453760, 51385466880, 175814737920, 366917713920, 2717245440000, 6782244618240, 22754631352320, 69918208819200

Offset: 0

Author

Hieronymus Fischer, Jul 08 2007

Keywords

• nonn

Comments

From R. J. Mathar, Dec 05 2012: (Start)

a(n) = b(n-1) because a(n) = Product_{k=1..n} ceiling(n/k) = Product_{k=1..n-1} ceiling(n/k) = n*Product_{k=2..n-1} ceiling(n/k) = Product_{k=1..1} (1+(n-1)/k)*Product_{k=2..n-1} ceiling(n/k).

The cases of the product are (i) k divides n but does not divide n-1, ceiling(n/k) = n/k = 1 + floor((n-1)/k), (ii) k does not divide n but divides n-1, ceiling(n/k) = 1 + (n-1)/k = 1 + floor((n-1)/k) and (iii) k divides neither n nor n-1, ceiling(n/k) = 1 + floor((n-1)/k).

In all cases, including k=1, a(n) = Product_{k=1..n-1} (1+floor((n-1)/k)) = Product_{k=1..n-1} floor(1+(n-1)/k) = b(n-1).

(End)

a(n) is the number of functions f:D->{1,2,..,n-1} where D is any subset of {1,2,..,n-1} and where f(x) == 0 (mod x) for every x in D. - Dennis P. Walsh, Nov 13 2015

Examples

			From _Paul D. Hanna_, Nov 26 2012: (Start)
Illustrate initial terms using formula involving the floor function []:
  a(1) = 1;
  a(2) = [2/1] = 2;
  a(3) = [3/1]*[4/2] = 6;
  a(4) = [4/1]*[5/2]*[6/3] = 16;
  a(5) = [5/1]*[5/2]*[7/3]*[8/4] = 60;
  a(6) = [6/1]*[7/2]*[8/3]*[9/4]*[10/5] = 144.
Illustrate another alternative generating method:
  a(1) = 1;
  a(2) = (2/1)^[1/1] = 2;
  a(3) = (2/1)^[2/1] * (3/2)^[2/2] = 6;
  a(4) = (2/1)^[3/1] * (3/2)^[3/2] * (4/3)^[3/3] = 16;
  a(5) = (2/1)^[4/1] * (3/2)^[4/2] * (4/3)^[4/3] * (5/4)^[4/4] = 60.
(End)
For n=3 the a(3)=6 functions f from subsets of {1,2} into {1,2} with f(x) == 0 (mod x) are the following: f=empty set (since null function vacuously holds), f={(1,1)}, f={(1,2)}, f={(2,2)}, f={(1,1),(2,2)}, and f={(1,2),(2,2)}. - _Dennis P. Walsh_, Nov 13 2015
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• D. P. Walsh, Notes on finite functions from subsets of {1,2,...,n} into {1,2,...,n}

Crossrefs

Cf. A010786, A131387, A075885, A131688, A308820, A092143.

Programs

Maple

a:= n-> mul(ceil(n/k), k=1..n):
seq(a(n), n=0..40); # Dennis P. Walsh, Nov 13 2015

Mathematica

Table[Product[Ceiling[n/k],{k,n}],{n,25}] (* Harvey P. Dale, Sep 18 2011 *)

PARI

a(n)=prod(k=1,n-1,floor((n+k-1)/k)) \\ Paul D. Hanna, Feb 01 2013

PARI

a(n)=prod(k=1,n-1,((k+1)/k)^floor((n-1)/k))
for(n=1,30,print1(a(n),", ")) \\ Paul D. Hanna, Feb 01 2013

Formula

a(n) = Product_{k=1..n} ceiling(n/k).

Formulas from Paul D. Hanna, Nov 26 2012: (Start)

a(n) = Product_{k=1..n-1} floor((n+k-1)/k) for n>1.

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

Limits: Let L = limit a(n+1)/a(n) = 3.51748725590236964939979369932386417..., then

(1) L = exp( Sum_{n>=1} log((n+1)/n) / n ) ;

(2) L = 2 * exp( Sum_{n>=1} (-1)^(n+1) * Sum_{k>=2} 1/(n*k^(n+1)) ) ;

(4) L = exp( Sum_{n>=1} (-1)^(n+1) * zeta(n+1)/n ) ;

(5) L = exp( Sum_{n>=1} log(n+1) / (n*(n+1)) ) = exp(c) where c = constant A131688.

Compare L to Alladi-Grinstead constant defined by A085291 and A085361.

(End)

a(n) = A308820(n)/A092143(n-1) for n > 0. - Ridouane Oudra, Sep 28 2024

Extensions

a(0)=1 prepended by Alois P. Heinz, Oct 30 2023