On the Set {π(kn) : k=1, 2, 3, ...}

摘要

An open conjecture of Z.-W. Sun states that for any integer n > 1 there is a positive integer k ≤ n such that π(kn) is prime, where π(x) denotes the number of primes not exceeding x. In this paper, we show that for any positive integer n the set {π(kn) : k=1, 2, 3, ...} contains infinitely many P_2-numbers which are products of at most two primes. We also prove that under the Bateman-Horn conjecture the set {π(4k) : k=1, 2, 3, ...} contains infinitely many primes.