a(n) and a(n)+4^k are primes at least for k=1,2,3,4,5.

Terms

a(0) =7a(1) =37a(2) =163a(3) =9157a(4) =9277a(5) =15667a(6) =53593a(7) =56893a(8) =111577a(9) =135193a(10) =137383a(11) =142543a(12) =305407a(13) =467527a(14) =470647a(15) =476023a(16) =480043a(17) =527377a(18) =607093a(19) =671353a(20) =761377a(21) =817147a(22) =885943a(23) =891643a(24) =904663a(25) =1080073a(26) =1116637a(27) =1140847a(28) =1172803a(29) =1233523