Question: Is there any formula for finding the $\operatorname{gcd}(\phi(n), n)$?

I'm not sure if this is a dumb question, but I couldn't find one myself and not on Wikipedia.

**EDIT**: To clarify what I'm trying to do:

I'm trying to solve another problem, where I have to plug the greatest common divisor into the Totient function again, and it would be fun if there was an expression for that so it maybe would simplify.