What are the most (and least) likely factors of a composite Mersenne number?
Suppose some number $M_p=2^p-1:p\in\text{prime}$ is a candidate for a Lucas Lehmer test. Is it possible to identify a set of the most-likely factors for pre-sieving before starting?
EDIT: A comment says factors of $M_p$ must be of the form $2kp+1$, so are there subsets of those which cannot be, or are less- or more-likely to be factors?