For integers $a,b,k$ with $a \geq 1$ and $k\geq 2$, consider the shifted geometric progression $n_i = ak^i + b$. I would like to understand the set of integers (prime or otherwise) that divide at least one of the $n_i$'s. More precisely, let
$$ D = D_{a,b,k} = \{ d \in \mathbb{N} \ : \ \text{there exists } i \geq 1 \text{ with } d \mid n_i \}, $$
$$ P = P_{a,b,k} = \{ p \ : \ p \text{ is prime and there exists } i \geq 1 \text{ with } p \mid n_i \}. $$
**Questions:**

Is it known the set $P$ has positive relative density inside the primes?

Barring a positive answer to 1., what can be said about the sizes of $D$ and $P$?

(It is certainly possible that $D$ has asymptotic density $0$, e.g. by letting $a,b$ and $k$ be squares. If the answer to the first question is positive then $\# D \cap [1,x] \geq \# P \cap [1,x] \gg x/\log x$. In the second question I'm hoping that perhaps a weaker asymptotic is true, such as $\# D \cap [1,x] \gg x^{1-\varepsilon}$.)

**Context:** It would follow from Artin's conjecture on primitive roots that $k$ is a primitive root for a positive proportion of the primes, and for each such prime $p$ there exists $i$ with $k^i \equiv -b/a$, whence $p \mid n_i$ (at least, as long as $p$ does not divide $ab$). Hence, if we believe Artin's conjecture then $P$ has positive density inside the primes.

One can also show by purely elementary methods that for each $C$ there exists $i$ such that $n_i$ has $>C$ prime divisors. Hence, $P$ is infinite. Here is a sketch: Assume, as we may do without loss of generality, that $b$ is coprime to $k$ (else, replace $a$ with $ak^{i_0}/d$, $b$ with $b/d$, and $i$ with $i-i_0$ where $d = \gcd(k^{i_0},b)$ for some large integer $i_0$). This assumption ensures that $n_i$ ($i \geq 1$) are coprime to $k$. Construct a sequence $i_j$ where $i_1$ is arbitrary, and $i_{j+1} = i_j + \varphi(n_{i_j}^2)$, so that $n_{i_{j+1}} \equiv n_{i_j} \bmod n_{i_j}^2$. This is set up so that $n_{i_{j+1}}/n_{i_j}$ is an integer coprime to $n_{i_j}$, and consequently $n_{i_j}$ has at least $j$ distinct prime factors.