TITLE: Are there infinitely many primes p such that both p-1 and p+1 have at most 3 prime factors, counted with multiplicity?
QUESTION [7 upvotes]: 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.

REPLY [11 votes]: At the present time, the best (general) result is due to James Maynard, who has proven that any admissible 3-tuple takes at most 7 prime factors infinitely often (see this paper).  In particular, (using a certain admissible 3-tuple related to what Robert Israel wrote) we can get that $n(n^2-1)$ has at most 11 prime factors infinitely often.
I don't know if this number can be improved for the specific case you are interested in.  However, the general consensus is that Dickson's conjecture is true, so both $p-1$ and $p+1$ should consist of exactly 3 prime factors, for infinitely many $p$.  But this conjecture is currently out of reach of any known techniques, and appears to be harder than the twin prime conjecture, since we must control admissible 3-tuples, but the twin prime problem deals with only admissible 2-tuples.