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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    $\begingroup$ @BenBarber No, he wants as many as possible of these multiples to constitute an interval $I\subset[n,2n]$. $\endgroup$ – Wolfgang Nov 30 '15 at 14:00
  • $\begingroup$ @Wolfgang Thank you, that was a particularly silly misreading of the question. I'll delete it to reduce clutter. $\endgroup$ – Ben Barber Nov 30 '15 at 14:09

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