This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A010786 #67 Feb 16 2025 08:32:32
%S A010786 1,1,2,3,8,10,36,42,128,216,600,660,3456,3744,9408,18900,61440,65280,
%T A010786 279936,295488,1152000,2116800,4878720,5100480,31850496,41472000,
%U A010786 93450240,163762560,568995840,589317120,3265920000,3374784000,11324620800,19269550080,42188636160
%N A010786 Floor-factorial numbers: a(n) = Product_{k=1..n} floor(n/k).
%C A010786 Product floor(n/1)*floor(n/2)*floor(n/3)*...*floor(n/n).
%C A010786 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 A010786 Reinhard Zumkeller, <a href="/A010786/b010786.txt">Table of n, a(n) for n = 0..1000</a>
%H A010786 Vaclav Kotesovec, <a href="/A010786/a010786_1.jpg">Graph - The asymptotic ratio (1000000 terms)</a>
%H A010786 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Alladi-GrinsteadConstant.html">Alladi-Grinstead Constant</a>
%H A010786 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A010786 a(n+1) = a(n)*A208449(n)/A208450(n). - _Reinhard Zumkeller_, Feb 26 2012
%F A010786 GCD(a(n), a(n+1)) = A208448(n). - _Reinhard Zumkeller_, Feb 26 2012
%F A010786 From _Vaclav Kotesovec_, Oct 03 2018: (Start)
%F A010786 log(a(n)) ~ c * (n - log(2*Pi*n)/2), where c = 0.7885...
%F A010786 Conjecture: c = A085361. (End)
%F A010786 From _Ridouane Oudra_, Jan 18 2025: (Start)
%F A010786 a(n) = Product_{k=1..n} ((k+1)/k)^floor(n/(k+1)).
%F A010786 a(n) = Product_{k=1..n} k^A075993(n, k).
%F A010786 a(n) = A092143(n)/f(n), where f(n) = Product_{k=1..n} ((floor(n/k)-1)!).
%F A010786 a(n) = A092143(n)/g(n), where g(n) = Product_{k=1..n} A377484(k).
%F A010786 a(n)/a(n-1) = A007955(n)/A377484(n). (End)
%e A010786 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 A010786 a := n -> mul( floor(n/k), k=1..n);
%t A010786 Table[Product[Floor[n/k],{k,n}],{n,40}] (* _Harvey P. Dale_, May 09 2017 *)
%o A010786 (Haskell)
%o A010786 a010786 n = product $ map (div n) [1..n]
%o A010786 -- _Reinhard Zumkeller_, Feb 26 2012
%o A010786 (PARI) vector(50, n, prod(k=1, n, n\k)) \\ _Michel Marcus_, Nov 10 2014
%o A010786 (Magma) [&*[n div i: i in [1..n]]: n in [1..35]]; // _Vincenzo Librandi_, Oct 03 2018
%Y A010786 Cf. A006218, A075885, A092143, A131385, A131387, A377484, A007955, A075993.
%K A010786 nonn,nice
%O A010786 0,3
%A A010786 _N. J. A. Sloane_, _Simon Plouffe_
%E A010786 More terms from _Hieronymus Fischer_, Jul 08 2007
%E A010786 Edited by _N. J. A. Sloane_, Jul 05 2008 at the suggestion of _Rick L. Shepherd_
%E A010786 a(0)=1 prepended by _Alois P. Heinz_, Oct 30 2023