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Let $\delta_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$. Is $\delta_1(n,m)$ unimodular for $m>n+1$ (i.e. increases until some $m$ then decreases thereafter)? For fixed $n$, where does $\delta_1(n,m)$ achieve its maximum?
Erdős proved that \[\delta_1(n,m) \ll \frac{1}{(\log n)^c}\] for all $m$, for some constant $c>0$. Sharper bounds (for various ranges of $n$ and $m$) were given by Ford [Fo08].

Stijn Cambie has calculated that unimodularity fails even for $n=2$ and $n=3$. For example, \[\delta_1(3,6)= 0.35\quad \delta_1(3,7)\approx 0.33\quad \delta_1(3,8)\approx 0.3619.\]

Furthermore, Cambie [Ca25] has shown that, for large $n$, the sequence $\delta_1(n,m)$ has infinitely many local maxima $m$.

See also [446].

Additional thanks to: Stijn Cambie