Problem Statement
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?
Categories:
Number Theory Divisors
Progress
This was asked by Erdős in 1986 at Oberwolfach.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].
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 fixed $n$, the sequence $\delta_1(n,m)$ has superpolynomially many local maxima $m$.
See also [446].
Source: erdosproblems.com/692 | Last verified: January 16, 2026