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A010786 Floor-factorial numbers: a(n) = Product_{k=1..n} floor(n/k). 26

%I #63 Oct 30 2023 18:20:16

%S 1,1,2,3,8,10,36,42,128,216,600,660,3456,3744,9408,18900,61440,65280,

%T 279936,295488,1152000,2116800,4878720,5100480,31850496,41472000,

%U 93450240,163762560,568995840,589317120,3265920000,3374784000,11324620800,19269550080,42188636160

%N Floor-factorial numbers: a(n) = Product_{k=1..n} floor(n/k).

%C Product floor(n/1)*floor(n/2)*floor(n/3)*...*floor(n/n).

%C a(n) is the number of functions f:[n]->[n] where f(x) is a multiple of x for all x in [n]. We note that there are floor[n/x] possible choices for each image of x under f. [_Dennis P. Walsh_, Nov 06 2014]

%H Reinhard Zumkeller, <a href="/A010786/b010786.txt">Table of n, a(n) for n = 0..1000</a>

%H Vaclav Kotesovec, <a href="/A010786/a010786_1.jpg">Graph - The asymptotic ratio (1000000 terms)</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Alladi-GrinsteadConstant.html">Alladi-Grinstead Constant</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%F a(n+1) = a(n)*A208449(n)/A208450(n). - _Reinhard Zumkeller_, Feb 26 2012

%F GCD(a(n), a(n+1)) = A208448(n). - _Reinhard Zumkeller_, Feb 26 2012

%F From _Vaclav Kotesovec_, Oct 03 2018: (Start)

%F log(a(n)) ~ c * (n - log(2*Pi*n)/2), where c = 0.7885...

%F Conjecture: c = A085361. (End)

%e For n=4 the a(4)=8 functions are given by the image sequences <1,2,3,4>, <1,4,3,4>, <2,2,3,4>, <2,4,3,4>, <3,2,3,4>, <3,4,3,4>, <4,2,3,4>, and <4,4,3,4>. [_Dennis P. Walsh_, Nov 06 2014]

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

%t Table[Product[Floor[n/k],{k,n}],{n,40}] (* _Harvey P. Dale_, May 09 2017 *)

%o (Haskell)

%o a010786 n = product $ map (div n) [1..n]

%o -- _Reinhard Zumkeller_, Feb 26 2012

%o (PARI) vector(50, n, prod(k=1, n, n\k)) \\ _Michel Marcus_, Nov 10 2014

%o (Magma) [&*[n div i: i in [1..n]]: n in [1..35]]; // _Vincenzo Librandi_, Oct 03 2018

%Y Cf. A006218, A075885, A092143, A131385, A131387.

%K nonn,nice

%O 0,3

%A _N. J. A. Sloane_, _Simon Plouffe_

%E More terms from _Hieronymus Fischer_, Jul 08 2007

%E Edited by _N. J. A. Sloane_, Jul 05 2008 at the suggestion of _Rick L. Shepherd_

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

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Last modified March 28 13:21 EDT 2024. Contains 371254 sequences. (Running on oeis4.)