Generalized Collatz: divide out by $2$’s and $3$'s, otherwise $5n+1?$

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?