It seems evident that for $a,b,c$ with $a>0$ and $b>1$ that there are only $o(x)$ primes of the form $a\cdot b^n+c$ with $n\le x.$ Has this been proven?
Hooley (Applications of Sieves to the Theory of Numbers) proves this for Cullen numbers, and apparently this proof generalizes to forms $n\cdot2^{n+a}+b$.
A reference would be great, if one can be found. Otherwise, I'll take what I can get!
It seems that the problem is entirely open, as seen by this answer on MathOverflow.
The difficulty is that the Cullen form gets equidistribution in all odd moduli from the $n$, while the general case has more restricted residue classes that are harder to control.