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

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

• Maybe it i helpful if you indicate what you are looking for: a way of finding some of these numbers with a computer, or a general categorization. – Reiner Martin Jun 29 '17 at 19:55
• By "power of sum divisor function", do you mean $\sigma_x(n)=\sum_{d\mid n} d^x$ ? – user228113 Jun 29 '17 at 20:01
• yes , this what i meant !!!!!! – zeraoulia rafik Jun 29 '17 at 20:06

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 $468=2^2\cdot 3^2\cdot 13.$