A Cullen prime is a Cullen number $C_m$ that is also a prime number. The first two Cullen primes are 3 and 393050634124102232869567034555427371542904833, corresponding to $m = 1$ and 141 respectively; other $m$ to give Cullen primes are 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, etc. listed in A005849 of Sloane's OEIS (those primes being too large to write here). None of these indexes are prime, and it is a matter of conjecture whether there exists a Cullen prime of the form $p2^p + 1$ with $p$ also prime. It is not even known whether there are infinitely many Cullen primes (a possibility despite their rarity).