Why is it important the role of $3n+1$ in Collatz conjecture? I mean, if we replace $3n+1$ by $5n+1$ it seems (numerically) that the modified statement of Collatz conjecture does not hold in this case.
So I assume that putting $3n+1$ is special in some sense, but which is the idea behind it? Are there any informal ideas that can make us think that the long time behaviour with $3n+1$ should be distinct than with $5n+1$, $7n+1$,...
Does anybody know what happens when we change $3$ by another odd number (proof or at least some kind of intuition)?
Thank you for your time.