These are the prime numbers of the form M_{n} = 2^{n}-1
Not every value of n gives a prime M_{n}, for instance 2^{4}-1 = 15 is obviously not prime.
n must be a prime number, but it is not sufficient : 2^{11}-1 = 2047 = 23 x 89

M_{n}=3, 7, 31, 127, 8191, ...
We don't know if there are infinitely many.
The Mersenne primes are related with even perfect numbers.