# Mersenne quasi-primes

This question comes out of this recent question (and is a comment to my answer to it, but should be of independent interest: What is known about "Mersenne numbers" (those of the form $2^n-1$) with few divisors (you can define few any way you want; I want it to be smaller than the expected number for a number of that size).

• It's Mersenne with an s. Nov 23 '11 at 18:16
• There's a Lenstra-Pomerance-Wagstaff heuristic for number of Mersenne primes < x, so maybe this can be modified to get what you want? For instance, part of the derivation of the mentioned heuristic involves the point of the probability of a random number of size $2^n-1$ being prime being about $1/(n \log 2)$ (but there are other things also), and just the other day, there was a thread on Math Overflow (mathoverflow.net/questions/35927/…) on asymptotic density of $k$-almost primes that might be useful here. Nov 24 '11 at 1:34