A131385 - OEIS

%I #48 Sep 29 2024 04:06:40

%S 1,1,2,6,16,60,144,672,1536,6480,19200,76032,165888,1048320,2257920,

%T 8294400,28311552,126904320,268738560,1470873600,3096576000,

%U 16094453760,51385466880,175814737920,366917713920,2717245440000,6782244618240,22754631352320,69918208819200

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

%C From _R. J. Mathar_, Dec 05 2012: (Start)

%C 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).

%C 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).

%C 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).

%C (End)

%C 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

%H G. C. Greubel, <a href="/A131385/b131385.txt">Table of n, a(n) for n = 0..1000</a>

%H D. P. Walsh, <a href="http://capone.mtsu.edu/dwalsh/FINITEF.pdf">Notes on finite functions from subsets of {1,2,...,n} into {1,2,...,n}</a>

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

%F Formulas from _Paul D. Hanna_, Nov 26 2012: (Start)

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

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

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

%F (1) L = 2 * exp( Sum_{n>=1} log((n+1)/n)) / n ) ;

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

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

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

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

%F (End)

%F a(n) = A308820(n)/A092143(n-1) for n > 0. - _Ridouane Oudra_, Sep 28 2024

%e From _Paul D. Hanna_, Nov 26 2012: (Start)

%e Illustrate initial terms using formula involving the floor function []:

%e a(1) = 1;

%e a(2) = [2/1] = 2;

%e a(3) = [3/1]*[4/2] = 6;

%e a(4) = [4/1]*[5/2]*[6/3] = 16;

%e a(5) = [5/1]*[5/2]*[7/3]*[8/4] = 60;

%e a(6) = [6/1]*[7/2]*[8/3]*[9/4]*[10/5] = 144.

%e Illustrate another alternative generating method:

%e a(1) = 1;

%e a(2) = (2/1)^[1/1] = 2;

%e a(3) = (2/1)^[2/1] * (3/2)^[2/2] = 6;

%e a(4) = (2/1)^[3/1] * (3/2)^[3/2] * (4/3)^[3/3] = 16;

%e a(5) = (2/1)^[4/1] * (3/2)^[4/2] * (4/3)^[4/3] * (5/4)^[4/4] = 60.

%e (End)

%e 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

%p a:= n-> mul(ceil(n/k), k=1..n):

%p seq(a(n), n=0..40); # _Dennis P. Walsh_, Nov 13 2015

%t Table[Product[Ceiling[n/k],{k,n}],{n,25}] (* _Harvey P. Dale_, Sep 18 2011 *)

%o (PARI) a(n)=prod(k=1,n-1,floor((n+k-1)/k)) \\ _Paul D. Hanna_, Feb 01 2013

%o (PARI) a(n)=prod(k=1,n-1,((k+1)/k)^floor((n-1)/k))

%o for(n=1,30,print1(a(n),", ")) \\ _Paul D. Hanna_, Feb 01 2013

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

%K nonn

%O 0,3

%A _Hieronymus Fischer_, Jul 08 2007

%E a(0)=1 prepended by _Alois P. Heinz_, Oct 30 2023