two semiprimes and a prime as three consecutive numbers

Question

Three consecutive numbers comprise two semiprimes and a prime. Examples are 21, 22, 23 and 157, 158, 159. Do you think such trios are endless?

Answer 1

Unless the prime $p$ is the largest of the three, it is the safe prime $2q+1$ associated with a Sophie Germain prime $q$, and it is currently unknown wether infinitely many Sophie Germain primes exist.

In the remaining case, we have $p=2q-1$ instead. As far as I know, such primes have no established name, but are also merely conjectured to be infinite in number.

Answer 2

This is not a complete answer, but it parses the problem, and is too long for a comment.

All primes other than $2,3$ have the form $6m\pm 1$. Since the question is about the infinite number of such trios, we need only concern ourselves with large values of $m$

In such trios, the prime number can never be the middle. If it were, one of the outside numbers would have the form $6m$ which for $m>1$ cannot be a semiprime.

Thus, the trios will be of the forms $6m-3,\ 6m-2,\ 6m-1$ or $6m+1,\ 6m+2,\ 6m+3$. In the first case, that requires $(2m-1,\ 3m-1)\in \mathbb P$ and in the second case that requires $(3m+1,\ 2m+1)\in \mathbb P$ for the trio to contain two semiprimes.