Erdős Problem #690

The problem

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.

Statement quoted from erdosproblems.com/690, maintained by Thomas Bloom. Original source: [Er79e]. Current status there: solved.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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