I created a new variant of Collatz Procedure. As always, when we encounter a even $x$, we apply $x \rightarrow x/2$. However, if we encounter a odd $x$, we have the freedom to choose any natural number $N≥2$ and apply $x \rightarrow x^N - 1$. It seems that there are values of $x$ where $1$ is not...

Recently i got interested at a Relaxation of Collatz Conjecture. This goes just like the normal Collatz procedure, except when $x$ is even, you have the choice of applying either $x \rightarrow 3x+1$ or $x \rightarrow x/2$. The idea of adding the ability to chose to Collatz procedure intrigues me...