From this article on Sum of Euler totient function we have the following tree:
We know that if $p$ is prime, $\phi(p) = p - 1$, that is, $\phi(11) = 11 - 1 = 10$.
It's obvious that the above tree has a main stem and the $\phi$ branches finally reach this stem. The important number to obtain is the power of $2$ at which the $\phi$ of a number reach to the stem. This would help to determine the $\phi(n)$ for arbitrary n, specifically if it becomes possible, it would lead to prime numbers identification.
Another important note can be the reverse scan of tree. As an example, consider the number $4$ in the main stem. The numbers ${5, 13, 11}$ reach to main stem on number $4$. What if we could obtain that list for arbitrary $2^n$?
Any solution to these problems?