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 $2^n1$) 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).
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1$\begingroup$ It's Mersenne with an s. $\endgroup$– Felipe VolochNov 23, 2011 at 18:16

$\begingroup$ There's a LenstraPomeranceWagstaff 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 $2^n1$ 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. $\endgroup$– Timothy FooNov 24, 2011 at 1:34
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