Let $\operatorname{ GPF}(n)$ be the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$.


Is there a way to prove that the sequence $a\_{n+1}=\operatorname{ GPF}(xa\_n+y)$, eventually enter a cycle for all positive integers $x,a\_0,y>0$?


Is there any set of positive integers $x,a\_0,y>0$ such that 
$a\_{n+1}=\operatorname{ GPF}(xa\_n+y)\operatorname{ LPF}(xa\_n+y)$ diverges?


Where $\operatorname{ LPF}(n)\geq2$ is the *least* prime factor of $n$.



The sequence $a_{n+1}=$ the greatest prime factor of $(xa_n+y)$
There is a paper on this problem, Mihai Caragiu, Recurrences based on the greatest prime factor function, JP J. Algebra Number Theory Appl. 19 (2010), no. 2, 155–163, MR2796479 (2012a:11010). The summary begins, 


We introduce and discuss a generalized ultimate periodicity conjecture for prime sequences $\lbrace q\_n\rbrace\_{n\ge0}$ in which every term $q\_n$ is recursively defined as the maximum element of the finite set $\lbrace P(A\_jq\_{n-1}+B\_j)\mid j=1,\dots,k\rbrace$, where $P(x)$ represents the greatest prime factor of $x$, while $A\_j$ and $B\_j$ are fixed positive integers for $1\le j\le k$. 


I haven't seen the paper, just the summary in Math Reviews. 


