There is known Catalan sequence :
$C_n=2^{C_{n-1}}-1$ , with $C_0=2$
I have noticed that following sequence produces prime numbers for the first four terms (I don't know if the fifth term is a prime number or not) :
$P_n=2^{P_{n-1}}-3$ , with $P_0=3$
Are there some similar prime number sequences of the form : $P_n=2^{P_{n-1}}-a$ ?