TITLE: How many primes $p$ such that $p-1$ is squarefree?
QUESTION [5 upvotes]: 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.

REPLY [3 votes]: There are infinitely many such primes, by [Mirsky, Theorem 2] below with $l=-1$ and $k=2$. This result says that the density of such primes in the set of all primes is about $1/3$, more precisely, Artin's constant: $$c_{Artin}:= \prod_{p \ prime} (1-\frac{1}{(p-1)p}) = 0.3739558136192\dots $$
This result was pointed out by Charles R. Greathouse IV in comments of  [OEIS].



References
[Mirsky] The number of representations of an integer as the sum of a prime and a $k$-free integer. Amer. Math. Monthly 56 (1949), 17--19.
[OEIS] Primes p such that p-1 is squarefree, https://oeis.org/A039787