Given $a_0$ be an positive integer, define
$$ a_{n+1} = \begin{cases} 8a_n, & \text{if $a_n$ is odd} \\ \lfloor a_n/3\rfloor, & \text{if $a_n$ is even} \end{cases}$$
Now form the sequence $(a_n)_{n\in \mathbb{N}}$ by performing this operation repeatedly.
Example: $a_0=11,(a_n)_{n\in\mathbb{N}}=(11,88,29,232,77,616,205,1640,546,182,60,20,6,2,0,0,...)$
1 Question: For every $a_0,(a_n)_{n\mathbb{N}}$ will eventually reach the number $0$?
The above claim is true for all $a_0$ up to $10^6$.
Keith Matthews enlarge conjecture to trajectories starting with a negative integer. Go through Keith Matthews programming to insure mapping, link
Extended Claim
Let $a_0$ be an positive integer and $k$ be any odd positive integer greater than $1$, define
$$ a_{n+1} = \begin{cases} (k^2-1)a_n, & \text{if $a_n$ is odd} \\ \lfloor a_n/k\rfloor, & \text{if $a_n$ is even} \end{cases}$$
Now form the sequence $(a_{n,k})_{n\in \mathbb{N}}$ by performing this operation repeatedly
Example $a_0=11,k=5,(a_{n,5})_{n\in\mathbb{N}}=(11,264,52,10,2,0,0,...)$
2 Question: For every $a_0$ and odd $k\ge 3$ the sequence $(a_{n,k})_{n\mathbb{N}}$ will eventually reach the number $0$?
Programming for extended claim by Keith Matthews, including negative values, link
Extended Collatz conjecture
Let $a_0$ be an positive integer and $k$ be any even positive integer, define
$$ a_{n+1} = \begin{cases} (k+1)a_n+1, & \text{if $a_n$ is odd} \\ \lceil a_n/k\rceil, & \text{if $a_n$ is even} \end{cases}$$
Now form the sequence $(a_{n,k})_{n\in \mathbb{N}}$ by performing this operation repeatedly
Example $a_0=5,k=4,(a_{n,k})_{n\in\mathbb{N}}=(5,26,7,36,9,46,12,3,16,4,1,...)$
again we can ask about, every sequence will eventually reach at the number $1$?
Programming for extended Collatz conjecture by Keith Matthews, including negative values, link
More on extended Collatz conjecture
First: Let $a_0,t$ be an positive integer and $k$ be any even positive integer, define
$$ a_{n+1} = \begin{cases} (k^t+k^{t-1})a_n+k^{t-1}, & \text{if $a_n$ is odd} \\ \lceil a_n/k\rceil, & \text{if $a_n$ is even} \end{cases}$$
Now form the sequence $(a_{n,k,t})_{n\in \mathbb{N}}$ by performing this operation repeatedly then sequence will eventually reach at the number $1$
Second: Let $a_0$ be an positive integer and $k$ be any even positive integer greater than $2$ and $t\in\mathbb{Z}_{\ge 2}$, define
$$ a_{n+1} = \begin{cases} (k^t+k^{t-2})a_n+k^{t-2}, & \text{if $a_n$ is odd} \\ \lceil a_n/k\rceil, & \text{if $a_n$ is even} \end{cases}$$
Now form the sequence $(a_{n,k,t})_{n\in \mathbb{N}}$ by performing this operation repeatedly then sequence will eventually reach at the number $1$
I again extend claim on odd $k$
First: Let $a_0$ be an positive integer and $k$ be any odd positive integer greater than $1$ and $t\in\mathbb{Z}_{\ge 2}$, define
$$ a_{n+1} = \begin{cases} (k^t-k^{t-2})a_n, & \text{if $a_n$ is odd} \\ \lfloor a_n/k\rfloor, & \text{if $a_n$ is even} \end{cases}$$
Then the sequence $(a_{n,k,t})_{n\in \mathbb{N}}$ will eventually reach at the number $0$
Second: Let $a_0$ be an positive integer and $k$ be any odd positive integer greater than $1$ and $t\in\mathbb{Z}_{\ge 1}$, define
$$ a_{n+1} = \begin{cases} (k^t-k^{t-1})a_n, & \text{if $a_n$ is odd} \\ \lfloor a_n/k\rfloor, & \text{if $a_n$ is even} \end{cases}$$
Then the sequence $(a_{n,k,t})_{n\in \mathbb{N}}$ will eventually reach at the number $0$