Prove that $\frac{2(\pi(p)-2)}{p-1} \leq 0.6$ for all prime numbers $p \geq 31$

Question

Let $f(p) = \frac{2(\pi(p)-2)}{p-1}$ where $\pi$ is the prime counting function, prove that $f(p) \leq 0.6$ for all prime numbers $p \geq 31$.

Below are some examples of $f(p)$.

Answer 1

Well, removing multiples of $2,3,5$ between $n+1$ and $n+30$ allows you to bound $\pi(n+30)\leq \pi(n)+8$ for $n>5$, and $\frac{16}{30}<0.6$. So it remains to check $2\pi(n)-2\leq0.6\times(n-1)$ for $31\leq n\leq 60$ explicitly.

Answer 2

Note that $\phi(30)=8$ so among $30$ consecutive integers only $8$ are coprime to $30$ and thus cannot be prime (unless $p=2$, $3$ or $5$). Therefore we have $\pi(n)-3 <\frac{8}{30}n$, can you finish from here?