On the density of some sets of primes, V

Abstract

In the present paper we derive an asymptotic formula for $\sum\limits_{p \leqslant x,r_{k}(p)=q_{\tau}}1$, where $k$ is a product of different odd primes, $q_{\tau}$ is the $\tau$-th consecutive prime and $r_{k}(p)$ the least prime $q$ such that $(\mathrm{ord}_{p} \ q, k) = 1$.