First term of sequences A065091, A073501, A102325-A102328.
These are the smallest p primes such that 1+p^n are p-smooth, i.e. with all prime-divisors less than p. The same holds for A102325-A102328. For the majority of p's, 1+p^n appear to be not p-smooth, i.e. have large enough greatest prime divisors.
a(8) > 4.5*10^6. - Giovanni Resta, May 02 2017