When is $\gcd(\sigma(n),\phi(n))$ is a prime number with $n$ a postive integer?

Question

Let $\phi$ Euler totient function and $\sigma$ is power of sum divisor function, I want to know if there is any standard basics in number theory show us :When is $\gcd(\sigma(n),\phi(n))$ is a prime number ? with $n$ is a postive integer

Answer 1

As helpful fact note that both functions $\phi$ and $\sigma$ are multiplicative. This implies that if the prime decomposition of $n$ is $p_1^{a_1}\cdot\ldots\cdot p_k^{a_k},$ then $$ \phi(n) = \phi(p_1^{a_1})\cdot\dots\cdot \phi(p_k^{a_k}), $$ and similarly for $\sigma.$ Next, note $\phi(p^a) = p^{a-1}(p-1)$ and $\sigma(p^a)=1+p+\cdots+p^a.$ This makes it easier to decide whether $\gcd(\sigma(n),\phi(n))$ is prime.

For example, if $p$ is any odd prime, then $$ \gcd(\sigma(p),\phi(p)) = \gcd(p-1,1+p) = 2, $$ which is prime. But there are other numbers, such as $6=2\cdot 3,$ or $10=2\cdot 5,$ or $468=2^2\cdot 3^2\cdot 13.$