*(This is meant as a comment to @JoshuaZ 's [answer](https://mathoverflow.net/a/473078).)*

Here is a list of found cycles for a lot of  $m \ne 3$ in the $m x+1$ - generalization. *(I've not done any analysis about divergent sequences here, I could prove divergent sequences only in generalizations, where —for instance— $3x+1$ and $5x+1$ are intertwined in the definition.)*

[![list of found cycles in Collatz-generalizations][1]][1] 


  [1]: https://i.sstatic.net/pzixvvAf.jpg