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 $k\ge 1$ is $d_k(p)$ unimodular in $p$? That is, it first increases in $p$ until its maximum then decreases.

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Fix a prime $p$. Let $S$ range over subsets of the primes (<p) of size $k-1$. The condition “the $k$th smallest (distinct) prime factor is $p$” is exactly:

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.