One funny thing about Mersenne primes is that, as a result of what you describe, they are exactly those primes whose binary representation consists of a prime number of ones!
The smallest Mersenne prime, three, is binary 11, while the next largest is seven (111), then 31 (11111), then 127 (1111111). The next candidate, 2047 (11111111111), is not prime.
The smallest Mersenne prime, three, is binary 11, while the next largest is seven (111), then 31 (11111), then 127 (1111111). The next candidate, 2047 (11111111111), is not prime.