Generalization of Mersenne Primes

Question

Besides Mersenne primes, are there any other known conditions for the natural numbers $n$ and $m$ under which the natural number ${n^m - n + 1}$ is "unusually likely" to be prime? By unusually likely, I mean that if you were to compute a set of $k$ numbers of this form, where the greatest one is, let's say, $M$, then there are more natural numbers in your set than the expected value for a randomly sampled set of $k$ natural numbers less than or equal to $M$, and this is true for all sufficiently large ${k \in \mathbb{N}}$. One candidate I had in mind is the condition that $n$ is prime, but briefly testing it for ${n = 3}$, it seems to work for sufficiently small $m$, but not for larger $m$ (even if you require $m$ to be prime as well). I must admit, I just tested some values to see how likely it seemed they were prime, and did not precisely calculate any probabilities or expected values. I would be interested in something that gives a noticeable difference in the probabilities though, ideally.