I found by accident the notion of arithmetic derivative. It goes as follows: if $p$ is prime then $p' = 1$ and it follows the usual Leibniz rule $$ (p q)' = p' q + p q'\quad\forall \;p,q \in\mathbb{N}\,. $$ My question regards the iterated action of this derivative on natural numbers, that is, I want to study the behaviour of $$ n^{(k)} \equiv n^{\overset{k}{\overbrace{\prime\ldots\prime}}}\,. $$ This was inspired by a question in this video (which is where I found out about this).
I am not a matematician so I don't have many tools to study this. So far I just did a table with Mathematica. It seems that powers of 2 tend to diverge to infinity (apart from $4$ which is a fixed point and $2$ which is prime). And I also observed that some numbers tend to infinity sharing the same trajectory, for example $$ 160^{(5)} = 180^{(5)} = 4834\,, $$ and diverge together from then onward. Another simple observation is the for any prime $p$, $(p^p)' = p^p$. So there are infinitely many fixed points.
Is there anything about this in the literature? Are there fixed points other than $p^p$? What is the rate of growth of $n^{(k)}$ when it doesn't reach zero? Is there a criterion to know in advance if a number is going to reach zero? Even a partial answer to any of these questions will be appreciated. I attach a log-log plot of the result of 30 iterations of the first $65536$ numbers. Red dots indicate powers of $2$.
