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

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

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 $f_k$?

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 $f_k$?

Given a natural integer $k$ we define $f_k:\mathbb N\longrightarrow\mathbb N$ by $f_k(x)=x/2$ if $x$ is even and by $f_k(x)=(3x+3^k)/2$ if $x$ is odd. For $k=0$ we get therefore 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 $f_k$?

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

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

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 $f_k$?