x->x*ceiling(x)

[Ed Pegg]

Mathematica

Table[{n, First[Flatten[Position[Map[Denominator, NestList[# Ceiling[#] &, 
(n + 1)/n, 20]], 1]]]}, {n, 1, 20}]

{{1, 1}, {2, 2}, {3, 3}, {4, 4}, {5, 19}, {6, 3}, {7, 4}, {8, 5}, {9, 7}, 
{10, 8}, {11, First[{}]}, {12, 5}, {13, 10}, {14, 4}, {15,5}, {16, 9}, 
{17, 7}, {18, 5}, {19, First[{}]}, {20, 12}}

I've carried out 12/11 and 20/19 to 25 terms, without the denominator
going to 1.  I looked at the Modulus of the numerators last night, trying
to see patterns that might allow for a reduction in size.  

So, you can view the sequence as 
{1, 2, 3, 4, 19, 3, 4, 5, 7, 8, ?, 5, 10, 4, 5, 9, 7, 5, ?, 12} 
or
{0, 1, 2, 3, 18, 2, 3, 4, 6, 7, ?, 4, 9, 3, 4, 8, 6, 4, ?, 11}
depending on whether you count the starting condition or not.

--Ed Pegg jr

> This led me to see what happens in the simplest case, when you
> start with the first rational with denominator n, namely (n+1)/n.
> 
> %I A073524
> %S A073524 0,1,2,3
> %N A073524 Number of steps to reach an integer starting with 
> (n+1)/n and using the map x -> x*ceiling(x), or -1 if no integer 
> is ever reached.
> %C A073524 Suggested by a question raised by James Tanton, email 
> Aug 28, 2002.
> %O A073524 1,3
> %K A073524 nonn,more,bref
> %e A073524 a(7) = 3 since 8/7 -> 16/7 -> 48/7 -> 48.
> %A A073524 njas, Aug 29 2002
> 
> what is a(5)?   watch:

a(5) is 19, my my count