Let $g(n)$ denote the largest $t$ such that there exist integers $2\leq a_1<a_2<\cdots <a_t <n$ such that $$P(a_1)>P(a_2)>\cdots >P(a_t)$$ where $P(m)$ is the greatest prime factor of $m$. Estimate $g(n)$.

Let $g(n)$ denote the largest $t$ such that there exist integers $2\leq a_1<a_2<\cdots <a_t <n$ such that $$P(a_1)>P(a_2)>\cdots >P(a_t)$$ where $P(m)$ is the greatest prime factor of $m$. Estimate $g(n)$.

Stijn Cambie has proved [Ca25b] $$g(n) \asymp \left(\frac{n}{\log n}\right)^{1/2}.$$ Cambie further asks whether there exists a constant $c$ such that $$g(n) \sim c \left(\frac{n}{\log n}\right)^{1/2}.$$ Cambie's proof shows that such a $c$ must satisfy $2\leq c\leq 2\sqrt{2}$.

The sequence $a_1<a_2<\cdots<a_t$ is packaged as a strictly monotone map a : Fin t → ℕ with $2\leq a_i<n$, the greatest prime factor $P$ is Nat.maxPrimeFac, and $g(n)$ is the supremum in ℕ of the achievable lengths $t$.