One direction of generalizing Collatz is, instead of dividing out by the smallest prime $2$, to divide out by the first two smallest primes $2, 3$ and then do $5n + 1$ instead of $3n + 1$. It seems that if we define the function
$f(n) = \begin{cases} \frac{n}{2}, & \text{if } 2 | n, \\ \frac{n}{3}, & \text{if } 2 \nmid n \text{ and } 3 | n, \\ 5n+1, &\text{otherwise,} \end{cases}$
then $f(n)$ falls fairly quickly into the $\{1, 3, 6\}$-loop for small values of $n$. For instance, the trajectory for $n = 5$ is $$5 \to 26 \to 13 \to 66 \to 33 \to 11 \to 56 \to 28 \to 14 \to 7 \to 36 \to 18 \to 9 \to 3 \to 1,$$ whereupon it cycles endlessly. In fact, in this loop one can readily see convergence will happen for all $n < 17$, and for $n = 17$, the cycle goes $$17 \to 86 \to 43 \to 216 \to ... \to 1.$$
It seems that this converges rather quickly, and a similar generalization dividing out by $2, 3, 5$ or else doing a $7n+1$ step would seem to converge even faster. For large values of $k$, the function dividing out by all primes $2$ through $p_k$ and otherwise doing a $(p_{k+1})n+1$ step would seem absurdly unlikely to enter a nontrivial cycle, much less escape to infinity.
Have these generalizations been studied? Are results known? Is convergence established for one or more of them, perhaps for $k$ quite large?