A lucky prime is a lucky number that is also a prime number. So, in a sense, such numbers are doubly “lucky” because they've survived two different sieves. The first few are 3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193, etc., listed in A031157 of Sloane's OEIS. It's known that there are infinitely many primes and it's known that there are infinitely many lucky numbers, but it's not known if there are infinitely many lucky primes.