A Complete Answer to Erd\H{o}s Problem 690

Let $d_k(p)$ denote the natural density of positive integers whose $k$-th smallest prime divisor is $p$. Erd\H{o}s asked whether, for each fixed $k$, the sequence $p\mapsto d_k(p)$ is unimodal as $p$ ranges over the primes. Cambie proved that unimodality holds for $1\le k\le3$ and verified non-unimodality for $4\le k\le20$. We prove that $p\mapsto d_k(p)$ is not unimodal for every $k\ge4$, completing the classification. An exact first-difference criterion reduces the problem to comparing a symmetric-polynomial ratio with prime gaps. Explicit estimates for prime-counting functions, certified finite computations, one certified large prime gap, one certified twin prime, and a uniform Chinese-remainder construction then produce, for every $k\ge4$, a strict descent followed by a later strict ascent.