Quote:
Originally Posted by jasong
I'm going to expose my ignorance here. Because the numbers are of the form
k*2^n+1,
and k is prime, does this not increase the chance that a number has a small factor like 3 or 5? I mean, with other k's there's an increased chance, an observable phenomenon, that one or more small primes are represented in the kvalue. In my opinion, the fact that it's always a prime k decreases the chance that a random nvalue will yield a prime.
Am I wrong?

Yes you are right. If we were only looking at numbers n==1 (mod 3) then there is an increased chance of finding a chance. A greater chance by 3/2 ie 1.5x more likely.
Using a program I calculated that if you sieve a bunch of numbers to 20M, then each candidate remaining is 30 times more likely to be prime compared to each before sieving.
edit: this is true in general, but if a series has special form of factors like mersenne numbers, then this is totally off. No one knows if prime cullen factors have a special form or not.