> **Given a prime $\,p\ne3$, is it always possible to find another prime q such
> that $\,\phi(q)=\phi(p\cdot2^{2n+1})\,$ for some $n\in\mathbb{N}$** ($\,\phi\,$ is the Euler's totient function)?

Some examples are:

$p=2\;\rightarrow\;q=5\;\;\;(n=1)$

$p=7\;\rightarrow\;q=97\;\;\;(n=2)$

$p=31\;\rightarrow\;q=7681\;\;\;(n=4)$

Many thanks.