x->x*ceiling(x)

[Richard Guy]

These are something like what I was calling
`pseudo-GPs' and may be connected with PV numbers.
I'm copying this message to him in the hope that
he can put me right (or wrong!)  R.

On Thu, 29 Aug 2002, N. J. A. Sloane wrote:

> Someone called James Tanton posted the following message:
> 
> I have a question
> that has been puzzling me for a while now. 
> 
> Let R be a rational greater than 1. Now consider the sequence whose first
> term is R, and any term thereafter is the previous term rounded up to the
> nearest integer (if it is not already an integer) and multiplied by R. 
> 
> For example, the fraction R=11/6 yields the sequence of fractions:
> 
> 11/6 , 11/3 , 22/3 , 44/3 , 55/2 , 154/3 , 286/3 , 176 , 968/3 , 3553/6 ,
> ...
> 
> Notice the appearance of an integer (176).
> 
> Does every rational R>1 eventually produce an integer somewhere within its
> associated sequence?
> Two comments:
> 
> 1) It is "very likely" that an integer eventually appears in the sequence
> associated to a rational R=c/d. We're looking for the appearance of
> fraction whose numerator is congruent to zero (mod d). The "round up"
> feature bounces things around so it is likely to eventually be the case.
> 
> 2) As a point of interest: It is known that two different rationals cannot
> produce the same sequence. 
> ----------------------------------
> 
> 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:
> 
> 
> > f(6/5);
>                                      12/5
> 
> > f(%);
>                                      36/5
> 
> > f(%);
>                                      288/5
> 
> > f(%);
>                                     16704/5
> 
> > f(%);
>                                   55808064/5
> 
> > f(%);
>                                622908012647232/5
> 
> > f(%);
>                         77602878444025201997703040704/5
> 
> > 
> 
> 
> > f(%);
>          1204441348559630271252918141028336694332989128001036771264/5
> 
> > f(%);
> 290135792424028156178425357986052529062710984863337179470336908191924417208\
> 
>     517059859206222048920739921330978585792/5
> 
> 
> and so on ?  would someone like to help?
> 
> NJAS
> 
> here of course f in Maple is
> f:=x->x*ceil(x);
>