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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 : $P_n=2^{P_{n-1}}-a$ ?

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In short, no. This is because there is no known $a$ such that we can prove $2^n -a$ will be prime infinitely often.

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