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An open conjecture of Z.-W. Sun states that for any integer $n>1$ there is apositive integer $k\le n$ such that $\pi(kn)$ is prime, where $\pi(x)$ denotesthe number of primes not exceeding $x$. In this paper, we show that for anypositive integer $n$ the set $\{\pi(kn):\ k=1,2,3,\ldots\}$ contains infinitelymany $P_2$-numbers which are products of at most two primes. We also prove thatunder the Bateman--Horn conjecture the set $\{\pi(4k):\ k=1,2,3,\ldots\}$contains infinitely many primes.

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