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Median factor of factorials
What is the order of $m(N)$, the median prime in the prime factorization of $N!$, as $N\to\infty$? For example, $m(6)=2$ because $6!=2^4\times3^2\times5$ and the median of $\{2,2,2,2,3,3,5\}$ is $2$.
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