What's the greatest common divisor of $\phi(n)$ and $n$, where $\phi(n)$ is the Euler Totient Function?

Question

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.

Answer 1

There's no known closed expression about what you're asking.

However we can do a little bit better than that if we know the factorization of the integer $ n \ = \ p_1^{k_1}...p_r^{k_r}$

$\phi(n) \ = \ p_1^{k_1}...p_r^{k_r}(\frac{p_1 - 1}{p_1})...(\frac{p_r - 1}{p_r}) \ = \ p_1^{k_1-1}...p_1^{k_r-1}(p_1-1)...(p_r-1)$

$\gcd(\phi(n), n)\ =\ \gcd (p_1^{k_1-1}...p_1^{k_r-1}(p_1-1)...(p_r-1),\ p_1^{k_1}...p_r^{k_r})\ =\\ p_1^{k_1-1}...p_1^{k_r-1}\gcd((p_1-1)...(p_r-1), p_1...p_r)$