On composite solutions of $\operatorname{rad}(n-1)=\operatorname{rad}(\varphi(n))$

Question

After I've read the statement of Lehmer's totient problem, see this Wikipedia, I wondered what about composite solutions $n>4$ of the equation $$\operatorname{rad}(n-1)=\operatorname{rad}(\varphi(n)),$$

where $\varphi(n)$ is Euler's totient function and with $\operatorname{rad}(n)$ we denote the radical of an integer, see in Wikipedia this definition. I don't know if this equation was in the literature.

Question. Is it know from the literature or your calculations if is there a composite number $n>4$ satisfying $$\operatorname{rad}(n-1)=\operatorname{rad}(\varphi(n))?$$ Many thanks.

There aren't examples for $4<n<50$. Additionally I've searched in The On-Line Encyclopedia of Integer Sequences the string rad(phi(n)).

Answer 1

If you write a computer program, you'll find lots of counterexamples. For example, the following values of n yield counterexamples: $$1729,2431,6601,9605,10585,12801,15211,30889$$