On the set $\{\pi (kn):k=1,2,3,\dots \}$

An open conjecture of Z.-W. Sun states that for any integer $n>1$ there is a positive integer $k\le n$ such that $\pi (kn)$ is prime, where $\pi (x)$ denotes the number of primes not exceeding $x$. In this paper, we show that for any positive integer $n$ the set $\{\pi (kn):k=1,2,3,\dots \}$ 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 $\{\pi (4k):k=1,2,3,\dots \}$ contains infinitely many primes.