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.

Source: erdosproblems.com/690

Number theory

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An accepted solution exists. The statement is false.

Solved; the site's label is SOLVED, decided in the thread on 2026-05-07 for Cambie's refereed theorem: unimodal for $k=1,2,3$ , not unimodal for $4\le k\le20$ , so the sequence is not unimodal for every fixed $k$ . The classification for every $k\ge4$ is the pending Wang–Crapis claim.

Problem 690

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