OPEN
This is open, and cannot be resolved with a finite computation.
Let $d_k(p)$ be the density of those integers whose $k$th smallest prime factor is $p$ (i.e. if $p_1<p_2<\cdots$ are the primes dividing $n$ then $p_k=p$).
For fixed $k\geq 1$ is $d_k(p)$ unimodular in $p$? That is, it first increases in $p$ until its maximum then decreases.
Erdős believes that this is not possible, but could not disprove it. He could show that $p_k$ is about $e^{e^k}$ for almost all $n$, but the maximal value of $d_k(p)$ is assumed for much smaller values of $p$, at\[p=e^{(1+o(1))k}.\]A similar question can be asked if we consider the density of integers whose $k$th smallest divisor is $d$. Erdős could show that this function is not unimodular.
Cambie
[Ca25] has shown that $d_k(p)$ is unimodular for $1\leq k\leq 3$ and is not unimodular for $4\leq k\leq 20$.
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #690, https://www.erdosproblems.com/690, accessed 2026-01-16
