*(this is meant as a comment to @JoshuaZ 's answer)*       

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 is 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