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



Prime solutions of $\sigma(p)/2$ equal a power of a prime number
Each fixed $n$ is a special case of Schinzel's "Hypothesis H" (also known as the Bateman-Horn conjecture), which conjectures that there are infinitely many pairs of primes $p,q$ related by your equation $p=2q^n-1$. (For $n=1$ this is a special case of the Hardy-Littlewood prime $k$-tuples conjecture.) So it's conjectured that there are infinitely many examples for each positive $n$, but it's not known even whether there are infinitely many among all $n$ together.


