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
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.
(End)
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
KEYWORD
nonn
AUTHOR
Hieronymus Fischer, Jul 08 2007
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Oct 30 2023
STATUS
approved