The sequence $a_{n+1}=$ the greatest prime factor of $(xa_n+y)$ 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 $n$.
 A: 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. 
A: Not a full answer, but one can construct arbitrary long increasing $a_n$.
Since the primes contain arbitrary long arithmetic progressions, one can construct arbitrary long increasing $a_n$ - set $a_0$ to the first prime in the progression, $x=1$ and $y$ the difference of the progression.
So $a_n = \operatorname{GPF}(a_0 + n y)$ and for $n$ term prime AP $a_n$ is an increasing sequence of primes.
