Erdős Problem 690

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.

The answer is no in general: Cambie [Ca25] has shown that $d_k(p)$ is unimodular for $1\leq k\leq 3$ and is not unimodular for $4\leq k\leq 20$.

The densities $d_k(p)$ exist (see erdos_690.variants.hasDensity), so the statement quantifies over any function d recording them.