RS-SMO 2022 Problem 4

Let $f(n)$ be number of numbers $x \in \{1,2,\cdots ,n\}$ , $n\in\mathbb{N}$ , such that $gcd(x, n)$ is either $1$ or prime. Prove $$\sum_{d|n} f(d) + \varphi(n) \geq 2n$$ For which $n$ does equality hold?