Composition of Euler’s $phi$ function

I am trying to show that the $phi(phi(n))$ is equal to the number of generators of $mathbb{Z}_n^*$ for all $n$ such that a generator exists.

Generators only exist for cyclic groups, and so I know that $n$ must be equal to $1,2,4,p^k$, or $2p^k$ where $p$ is an odd prime and $k>0$.

I also know that the number of elements of order $n$ (the number of generators) will be equal to $phi(n)$

From here though I am unsure where to go. Any help would be greatly appreciated.