In the famous proof of infinitely many primes, one assumes there are finitely many and then defines $n=p_1p_2\cdots p_k$. Now when we consider $n+1$, it is a possibility that $n+1$ could be a prime. I would like to know how many such number are there where $n+1$ is a prime and study the behavior of all the numbers of this form. Eg: $3, 7, 11$, etc.
I tried re-framing the question to see if it help in any way: "How many primes $p$ exist such that $p-1$ is square free?" I was not able to come to any solutions.