Fouvry showed that the relative density is positive of primes $p$ for which the largest prime factor of $p+a$ is $\ge p^{\alpha}$ for $\alpha > \approx .6687$.
Etienne Fouvry, Th ́eoreme de Brun-Titchmarsh; application au th ́eoreme de Fermat, Invent.
Math 79 (1985), 383–407. MR0778134 (86g:11052)

