A216666 and A216668 sequences
I recently added 2 new sequences to OEIS.
A216666 : Smallest integer b such that (b+1)^p - b^p is prime for all primes p <= prime(n).
The 8 first terms are 1, 1, 1, 1, 6524, 809652228, 30717523794, 55779743835 and for p=23, a solution is > 336831150306.
A216668 : Smallest integer b > 1 such that for all primes p <= prime(n), (b+1)^p - b^p is prime if and only if 2^p-1 is prime.
The 11 first terms are 2, 2, 2, 296, 296, 369719, 457578, 43410300320, 43410300320, 43410300320, 288975248496 (288975248496 is not yet published at OEIS since it requires approval).
For p=61, a solution is > 2147176502688.
I wonder if lows can roughly express the growth of these sequences.
Maybe something like A(n+1) ~ A(n)^constant