I found this problem in a old number theory test about arithmetic functions. The problem says that a number $n \in N$ is "perfectly crazy" if $$\phi(n)^{\sigma(n)^{\tau(n)}}=n^2,$$ and, as an example, 1 is "perfectly crazy" and I need to find all "perfectly crazy" numbers. I believe there aren't any prime numbers perfectly crazy after some tests and I don't know how to prove that there aren't any at all.
$\phi(n)$ being Euler totient function
$\sigma(n)$ is the sum of positive divisors of $n$
$\tau(n)$ is the number of positive divisors of $n$