[math-fun] stupid number theory problem

Consider the sequence of integers generated by: Given a prime p, Ap_0 = p Ap_(n+1) = 2*Ap_n + 1 This sequence *terminates* when Ap_n *isn't* prime. Finally, define the function F(p), p prime, as the *length* of this sequence Ap_n. Informally, F(p) is the length of an all-prime sequence consisting of p, 2p+1, 2(2p+1)+1, ... These sequences of primes seem to be common enough, that I'm guessing that for *every* n, there exists a prime p such that F(p)=n. Here's the real kicker: there probably isn't a prime p s.t. F(p)=oo, is there ?