Groups of prime number-1

There are several groups of prime number. I will show you some.
1. Twin prime numbers: when p and p+2 are both prime numbers, human call them 'twin prime numbers.'
ex)  (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43)
2. Cousin prime numbers: when p and p+4 are both prime numbers, human call them 'cousin prime numbers.'
ex) (3, 7), (7, 11), (13, 17), (19, 23), (37, 41)
3. Mersenne prime numbers: when 2^a is a prime number, human call the numder 'Mersenne prime numbers.'
ex) 3, 7, 31, 127, 511, 2047