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.
Disclaimer: The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.
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 2026-04-11
