Problem 408

Erdős Problem #408: declared status 'open'. Formalized: no. Let $\phi(n)$ be the Euler totient function and $\phi_k(n)$ be the iterated $\phi$ function, so that $\phi_1(n)=\phi(n)$ and $\phi_k(n)=\phi(\phi_{k-1}(n))$. Let$$f(n) = \min \{ k : \phi_k(n)=1\}.$$Does $f(n)/\log n$ have a distribution function? Is $f(n)/\log n$ almost always constant? What can be said about the largest prime factor of $\phi_k(n)$ when, say, $k=\log\log n$? Prize: no. OEIS: A049108. Tags: iterated functions, number theory.