The $3n + 1$ conjecture states that if you take any natural number $n_j$, and if it is even then set $n_{j+1} = n_j/2$, otherwise set $n_{j+1} = 3n_j + 1$, then no matter what natural number $n_0$ you start with, eventually you will reach $n_j = 1$ for some $j$. It is still an open problem and believed to be very hard.
Has anybody studied this problem with different and/or more primes than just $3$ and $2$? For example, let's say that if a number $n$ is divisible by 3 then we divide by 3, and if it's divisible by 2 then we divide by 2, otherwise we form the number $7n + 1$. Are there any variants like this where the answer is known whether you will always reach $1$ no matter what natural number you start with?
Yes, many variants are known. If you're making up such a variant, it's not hard to construct one where you get a nontrivial cycle. On the other hand, the general problem is undecidable.