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A131385
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Product ceiling(n/1)*ceiling(n/2)*ceiling(n/3)*...*ceiling(n/n) (the 'ceiling factorial').
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10
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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
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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a(n) = Product_{k=1..n} ceiling(n/k).
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)
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EXAMPLE
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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
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MAPLE
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a:= n-> mul(ceil(n/k), k=1..n):
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MATHEMATICA
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Table[Product[Ceiling[n/k], {k, n}], {n, 25}] (* Harvey P. Dale, Sep 18 2011 *)
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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