For fixed $k\geq 1$ is $d_k(p)$ unimodular in $p$? That is, it first increases in $p$ until its maximum then decreases.
For fixed $k\geq 1$ is $d_k(p)$ unimodular in $p$? That is, it first increases in $p$ until its maximum then decreases.
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 $4\leq k\leq 20$.