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If I have a number $n$ and I count all the prime numbers below $n$, for how many numbers will there be more primes below $n$ than half of $n$?

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Hint: Taking the fact that $1$ is not usually counted a prime, how many prime even numbers are there below $n$?

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  • $\begingroup$ Approximately $\frac{n}{\ln(n)}$ which is almost always smaller than $\frac{n}{2}$, but there are a few numbers for which the condition does hold (namely $3$, $5$ and $7$)... right? $\endgroup$ – barak manos Jun 13 '15 at 13:47
  • $\begingroup$ @barakmanos approximately half of the numbers below $n$ are even. If $1$ is not counted as prime the number of primes below $3$ is $1$ and half of $3$ is $1.5$. $\endgroup$ – Mark Bennet Jun 13 '15 at 13:50
  • $\begingroup$ Oh, I took "below" as inclusive, thanks... $\endgroup$ – barak manos Jun 13 '15 at 13:58

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