Given $n$, what is the largest set of consecutive integers in $[n,2n]$ can we have so that each integer is divisible by a distinct element from $[\log n,2\log n]$ (no partiular order)? So apriori I am asking if we can achieve the maximum of $\log n$ consecutive integers.
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1$\begingroup$ @BenBarber No, he wants as many as possible of these multiples to constitute an interval $I\subset[n,2n]$. $\endgroup$– WolfgangCommented Nov 30, 2015 at 14:00
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$\begingroup$ @Wolfgang Thank you, that was a particularly silly misreading of the question. I'll delete it to reduce clutter. $\endgroup$– Ben BarberCommented Nov 30, 2015 at 14:09
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