The sequence ${\displaystyle{M_n:=2^{p_n}-1}}$, where ${\displaystyle{n\gt0}}$ and ${p_n}$ is the ${\displaystyle{n}^{th}}$ prime number, is commonly known as the Mersenne numbers (not to be confused with the Mersenne primes, which also requires the number itself to be prime). They have the property that no member of this sequence is divisible by ${2}$ (all Mersenne numbers are also odd numbers). In general, the sequences ${\displaystyle{a_{m,n}:=p_m^{p_n}-p_{m-1}\#}}$, where ${\displaystyle{m,n>0}}$ and ${\displaystyle{p_m\#:=\prod_{i=1}^mp_i}}$ is the ${\displaystyle{m^{th}}}$ primorial number, contain no members divisible by the first ${\displaystyle{m}}$ primes. The Mersenne numbers are a special case where ${\displaystyle{m=1}}$.
Do these sequences have a name? Do any important properties of Mersenne numbers besides the indivisibility by the first ${\displaystyle{m}}$ primes generalize to these related sequences?
Let's broaden the family of sequences somewhat. Let $b,c$ be integers, with $b\gt 1$, and consider the integer sequence for $k = 0,1,2,\ldots$:
$$ s_k = b^k - c $$
The case you ask about is when $b=p$ is prime and $c = (p-1)\#$ is a primorial number, and you restricted attention to the subsequence of prime exponents $k$.
Let's call these exact powers with a fixed offset. Negative as well as positive values of the offset constant $c$ are allowed, so these sequences include as subsequences Fermat numbers as well as Mersenne numbers. Some less well-known sequences in this family are the Cunningham numbers:
$$ b^n \pm 1 $$
and the so-called Crandall primes (after Richard Crandall's U.S. Patent 5,159,632, although there is prior art by Bender and Castagnoli):
$$ 2^q - c \;\;\text{ for small odd } c $$
The latter were studied as providing a richer supply of primes (than the few dozen Mersenne primes) for use as prime moduli of elliptic curve cryptosystems.
The key feature of these sequences is that they result from iteration of a first-degree univariate polynomial:
$$ s_{k+1} = b s_k + (b-1)c $$
This leads to treatment by arithmetic dynamics. I'll illustrate some of the ideas by taking the example $b=3$ and $c=2$ of your constructions.