Given $k \in \mathbb N$, we define $f\_k: \mathbb N \longrightarrow \mathbb N$ by


$$ f\_k(x) = \begin{cases} \,\quad\dfrac{x}2 &\text{ if } x \text{ is even} \\\\ \dfrac{3x+3^k}{2} & \text{ if } x \text{ is odd} \end{cases} $$


For $k=0$, we have the function of the infamous Collatz conjecture.


*Does there exist for some $k\geq 0$ a strictly positive natural integer $N\_k$ whose iterates under $f\_k$ do not end up in the
trivial cycle $\lbrace 3^k,2\cdot 3^k\rbrace$ of $f\_k$?*



A mutation of the Collatz disease
Maybe it is worth to write my comments into an answer:


1. The question is equivalent to asking whether all elements of the form $n/3^k$ end up in the collatz cycle.
2. It is still an open question whether there is an integer $n$ where the collatz iterates go to infinity
3. It is still an open question whether there are other integral positive Collatz cycles
4. Iterating with starting value $n/3^k$ for some $k$ and $n\neq 0$, we always end up with an integer after finitely many steps. Each $x\mapsto 3x+1$ decreases the exponent of $3$ in the denominator by one. So the question is equivalent to asking whether the conjectures 2,3 hold.


