Suppose that someone proves that there exist a sequence (if that is not already known) of the form $a^n+b$ where $a \in \mathbb N \setminus \{1\}$ and $b \in \mathbb Z$ which has an infinite number of primes as its values.

Now let us define the set $S_k=\{1 \leq i \leq k: a^i+b\in \mathbb P\}$. Let us denote with $|S_k|$ the cardinality of $S_k$.

Do we have $\lim_{k \to \infty} \dfrac {|S_k|}{k} =0$?

In other words, even if there is no concrete example of the sequence of the form $a^n+b$ which has an infinite number of primes as its values can it be proven that the density of primes in such sequences always equals zero?