This is a subsequence of primes $p$ for which $p^2-1$ has at most 6 prime divisors counted with multiplicity.

This sequence described in the question is the sequence A079153 in OEIS.

I could not find references on the first sequence, but I found a mention of the second sequence in another MO question: Number of prime factors of the order of a finite non-abelian simple group with a reference to a survey article by Solomon from 2001: are there infinitely many primes $p$ such that $|PSL(2,p)|=(p-1)*p*(p+1)/2$ is a product of six prime factors? In the survey article it is said this problem is akin to the twin prime conjecture, but no references are given.