It is said that there are infinitely many odd $k$ that yield composite numbers of the form $k2^n+1$ for all $n$ (https://en.wikipedia.org/wiki/Sierpinski_number). These are called Sierpinski numbers.
My question is whether it is known that there are infinitely many $k$ that yield at least one prime number of the form $k2^n+1$ for some $n$. If so, are there any classes of numbers that we know for sure are not Sierpinski numbers (without having to brute-force verify that there is a prime in the sequence like we did with 10223).