A lucky prime is a number that is a prime number and a lucky number. The first few lucky primes are

3 7 13 31 37 43 67 73 79 127 151 163 193 211 223 241 283 307 331 349 367 409 421 433 463 487 541 577 601 613 619 631 643 673 727 739 769 787 823 883 937 991 997 1009

Twin lucky primes occur less often than twin primes in general, but in a similar proportion.