Maybe, This question is stupid.But, I want to ask.Because, really I dont Know answer.This problem may be similar to others. My Question is:
$$ f(n) = \begin{cases} Pn±Q & \text {if $n$ is odd} \\ \frac{n}{2} & \text {if $n$ is even} \end{cases} ,$$ and we can find such $k$ $$f^{k}(n)=1$$ Here $P,Q\in \mathbb{N}$
For Example: We know counter examples, for $"3n-5","3n-1",3n+5"$ problems. So that $f^{k}(n)≠1$
Give me such a problem that we do not have a counter-example,(In shortly $f^{k}(n)=1$)
If the example you give is $"3n ± Q"$, and "$Q≠1"$ it was very good.
(Please, edit or improve question for me,because Unfortunately, I'm not as knowledgeable as you.) Thanks so much!
Surely the article "On the '3x+1'-Problem" of R.E.Crandall of 1978 fits your question well (it is online you can find it). Here is a screenshot of a part of that article dealing with the general $qx+r$-problem-variant:
source: MATHEMATICS OF COMPUTATION, VOLUME 32, NUMBER 144, Oct 1978
As an interesting sidenote: I've found a second cycle with the $q=181$ - problem. And also the $q=3511$ (having a similar property as $q=1093$ being a wieferich prime) should have been looked at.
But I think there was no substantial progress over this material of 1978... (it should then also be mentioned in the wikipedia, btw.)