Divisors of shifted geometric progressions 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.
 A: This does not fully answer the question, but gives several links to some related literature.
Most papers below study if the order of an element mod $p$ is odd or even,
(or more general). Therefore the links below study the cases $a=1, b=1$ or $b=-1$ in your notation.
Shparlinski's paper is more general, but primarily studies related sequences with more than 2 summands.
a) The following two papers by Hasse give at least some special cases
such as prime divisors of the sequence $k^n+1$, ($k$ is fixed, $n \in \mathbf{N}$).
In particular Hasse proved that the Dirichlet density of primes dividing an integer of the type $2^n+1$ is
$17/24$.
Note: if $p|(2^n+1)$, then $2^n\equiv -1 \bmod p$, and the order of $2$ modulo $p$ is even.
H. HASSE, Über die Dichte der Primzahlen p, für die eine vorgegebene ganzrationale Zahl
$a\neq 0$ von gerader bzw. ungerader Ordnung mod p ist. Math. Ann. 166 (1966), 19-23.
http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002296616
H. HASSE, Über die Dichte der Primzahlen p, für die eine vorgegebene rationale Zahl $a\neq 0$
von durch eine vorgegebene Primzahl $l \neq 2$ teilbarer bzw. unteilbarer Ordnung mod $p$ ist.
Math. Ann. 162 (1965), 74-76
http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002295253
b) Odoni proved corresponding results for natural density.
Journal of Number Theory
Volume 13, Issue 3, August 1981, Pages 303-319
A conjecture of Krishnamurthy on decimal periods and some allied problems,
https://www.sciencedirect.com/science/article/pii/0022314X81900160
In some cases he got positive relative prime density, in other cases 0-density.
c) On RH (related to Artin's conjecture
Stephens, P. J.
Prime divisors of second-order linear recurrences. I.
J. Number Theory 8 (1976), no. 3, 313–332.
d)
Shparlinski, Igor E.
Prime divisors of sparse integers.
Period. Math. Hungar. 46 (2003), no. 2, 215–222.
https://link.springer.com/article/10.1023%2FA%3A1025996312037
e) Hooley's book "Application of sieve methods to the theory of numbers"  has some information on prime factors of $2^n+b$.
f) Prime divisors of certain recurrent sequences have been studied e.g. by Ballot, and Moree.
Finally, I believe that the case $|b| \neq 1$ is more difficult,
as there is less algebraic structure (such as order of an element).
