All primes q>3 can be written on the form q=2*k*p + 1 for some k>0 and some prime p, and many primes q can be written like that in several different ways.

The number of different ways is the number of the distinct prime factors of (q-1)/2. Each distinct prime factor can be the prime p in k*p and then (q-1)/(2*p) is the k value.

Not all primes q=2kp+1 is the factor of a Mersenne number with prime exponent p, and those that are a factor, can only be the factor of 1 Mersenne number.

To be a factor the prime q (mod 120) must be equal to one of these 16 values:

1 7 17 23 31 41 47 49 71 73 79 89 97 103 113 119

Close to half of all primes satisfies this requirement but not all of those are factors. I was curious how many of those are Mersenne factors. I checked all factors in the GIMPS database < 10

^{9}, which is all there is < 10

^{9} since the smallest factor outside GIMPS range would be roughly 2*10

^{9}. 2kp+1 with k=1 and p>10

^{9}.

Here is the data, the primes%120 column is the number of primes satisfying the mod 120 requirements. I included the 6 Mersenne primes: M3=7, M5=31, M7=127, M13=8191, M17=131071, M19=524287 in the Mersenne factors column since they are really their own factors.

Not surprisingly the ratio of Mersenne factors to primes goes down with size. The percentages are the ratio of Mersenne factors to the primes%120 column.

Code:

n Total primes primes%120 Mersenne factors
10^{2} 25 11 4 36.36%
10^{3} 168 80 20 25.00%
10^{4} 1229 603 103 17.08%
10^{5} 9592 4783 583 12.19%
10^{6} 78498 39221 3842 9.80%
10^{7} 664579 332213 26556 7.99%
10^{8} 5761455 2880376 196645 6.83%
10^{9} 50847534 25422922 1511498 5.95%