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Theorem: A prime p is in the sequence iff p is a Fermat prime.
Proof: If p=2^2^n+1 is prime (Fermat prime) then phi(p)=2^2^n=2* phi(2^2^n)=2*phi(p-1), so p is in the sequence. Now if p is a prime term of the sequence then phi(p)=2*phi(p-1) so p-1=2*phi(p-1) and we deduce that p-1=2^m hence p is a Fermat prime.