(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 in generalizations, where -for instance- $3x+1$ and $5x+1$ are intertwined in the definition)
