The Hermite-grand-conjecture implies that f(k)=(2^(2^5^11^(7k+1))+1)/3 is prime for all natural numbers $k$.

Is there any explicit formula that has so far been proven to produce primes for all natural numbers?

If not, is there under some reasonable restriction of closed-form formula a possible non-constructive proof that there exist (or does not exist under even more restrictive conditions) a finite formula that produces primes for all natural integers?

If not, is there any formula that has been proven to output primes with a frequency approaching 1 for input naturals k approaching infinity?