Thanks to S. Carnahan for suggestion:

Question: I would like to know if there are results in the literature concerning the prime numbers of the form $$ p = 2q^n-1 $$ where $$ q $$ is an odd prime number.

Moreover, conjectural asymptotics are of special interest.

Furthermore, there seems not to be an entry in the OEIS about these prime numbers.

First terms are: $$ 5,13,17,37,53,61,73,97,\dots $$