Sun, Zhi-Wei; Zhao, Lilu On the set \(\{\pi(kn):k=1,2,3,\dots\}\). (English) Zbl 1461.11125 J. Comb. Number Theory 11, No. 2, 97-102 (2019). Summary: 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. MSC: 11N05 Distribution of primes 05A15 Exact enumeration problems, generating functions 11A07 Congruences; primitive roots; residue systems 11B99 Sequences and sets Keywords:prime-counting function; \(P_2\)-numbers; residue classes PDFBibTeX XMLCite \textit{Z.-W. Sun} and \textit{L. Zhao}, J. Comb. Number Theory 11, No. 2, 97--102 (2019; Zbl 1461.11125) Full Text: arXiv Online Encyclopedia of Integer Sequences: a(n) = |{0 < k < n: pi(k*n) is prime}|, where pi(.) is given by A000720.