What facts are known about the hypothetical smallest divergent integer in the collatz conjecture? If there is a divergent integer in the collatz conjecture then there must be a smallest divergent number by the WOP. We can observe some properties of this number such as it must be odd because if it was even then the next number in the series $n/2$ is smaller which is a contradiction. Are there any other known facts about the smallest number?
Such a number, call it $k$, cannot be congruent to $2 \;(\bmod\; 3)$, because otherwise it is possible to create a new counterexample.

For example, consider $\frac{2k-1}{3}$.

$$\frac{2k-1}{3}&lt;k$$

For $k &gt; -1$.

As $\frac{2k-1}{3}$ is odd, we iterate to $2k$. Being even, $2k$ goes to $k$.
Therefore, not only is $k$ a counterexample, so is $\frac{2k-1}{3}$. As it is lesser, this contradicts, $k$ as the least counter-example.

Therefore, $\frac{2k-1}{3}$ cannot exist. This happens when $k \not\equiv 2 \;(\bmod\; n)$. 

This logic passes the "$3n-1$" test, as if we take $7$, a known counter-example, we get $5=\frac{2\times7+1}{3}$, and $5$ is also a known counterexample.
Such a number must be of the form $4n-1$ since if it was of the form $4n+1$ you would lead to $3n+1$ which is smaller.