Is $\pi(n)=\left\lfloor \frac{n}{\log n} \right\rfloor$ for infinitely many $n$? If so, are there any conditions that a set or progression contains infinitely many such $n$'s? Do they have a distribution or any properties?
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Welcome to MSE. Please read this text about how to ask a good question. – José Carlos Santos Jul 26 '20 at 09:18
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@Vlad In the Non-asymptotic bounds on the prime-counting function section of Wikipedia's "Prime number theorem" article, consider the part showing the inequalities of Pierre Dusart, especially the first one (i.e., on the left). – John Omielan Jul 26 '20 at 09:43
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1And when you've digested that link, Vlad, maybe you'll be able to write up and post an answer to your own question! – Gerry Myerson Jul 26 '20 at 10:10
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@Vlad FYI, you may wish to read the closely related post of When does $\pi(x)$ cross $x/\log(x)$?. – John Omielan Jul 26 '20 at 21:38
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Something related to what you want can be found in this paper. Here I prove that for infinitely many $n$ $$\pi(n)=\frac{n}{\lfloor\ln{n}-1/2\rfloor}$$ holds true.
Konstantinos Gaitanas
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