Let a Collatz sequence of a natural number $x$ as follow:
$$\{x\}: \begin{cases} x_0=x\\ x_{n+1} = \begin{cases} \frac{3x_n+1}{2} \text{ if $x_n$ is odd}\\ \frac{x_n}{2} \text{ if $x_n$ is even} \end{cases} \quad\forall n\in\mathbb{N}^* \end{cases}$$
The sequence is said to be convergent iff there exists k such that $x_k=1$. However for the purpose of finding the smallest $x$ such that $\{x\}$ doesn't converge (if there's any), we also say the sequence converges iff there exists k such that $x_k<x$.
Let $f(x,n)$ be the amount of odd numbers in the sequence $\{x_0,x_1,...,x_n\}$. Find a function $g(n)$ such that if $f(x,n)\leq g(n)$, $\{x\}$ converges.
I believe such inequality must exist because if there are little enough odd numbers in the $\{x_0,...,x_n\}$ sequence, the sequence tends to decrease regardless of where the odd numbers lie in the sequence, and so there will exist $k\leq n$ such that $x_k<x$. However I don't know if anyone had found one.
In case of such $g(n)$ is found, function $h(n)$ is said to be better than $g(n)$ if $h(n)$ sastisfies the property that $g(n)$ has and $h(n)>g(n)$ for all large enough $n$. Then what is the current "best" function?