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On the set \(\{\pi(kn):k=1,2,3,\dots\}\). (English) Zbl 1461.11125

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
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