A131385 - OEIS

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

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OFFSET

0,3

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

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}

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 = 2 * 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

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A068787 A073959 A006820 * A027742 A324062 A033301