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.