It is a well-known problem, known as Collatz problem, to determine whether iteratively applying the map $f(x)=\frac{x}{2}\text{ if $x$ is even and }f(x)=3x+1\text{ otherwise}$ on a positive integer $n$ always will give us $1$. One possibility for which this could not be true is if the sequence $(f^k(n))_{k\in\Bbb N}$ was diverging to infinity. There are no such $n$, and for what I know, it is believed there are none.
Looking at this Math SE question I was mildly surprised to find out that if we replace $3$ by $5$, there are (apparently) no known trajectories diverging to infinity.
This let me to the following question:
Is there an odd integer $a$ and an integer $n$ such that it is known that iterating the map $f(x)=\frac{x}{2}\text{ if $x$ is even and }f(x)=ax+1\text{ otherwise}$ gives a sequence diverging to infinity?
Note that allowing $a$ even would make the problem trivial. I was considering extending this question to more general functions, but I couldn't fina a formulation which would sieve out trivial cases. Let me know in the comments if you know of an example of a diverging sequence which (in your opinion) counts as non-trivial.
Thanks in advance.