SOLVED (LEAN)
This has been resolved in some other way than a proof or disproof, and that resolution verified in Lean.
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 $2\leq c\leq 2\sqrt{2}$.
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Additional thanks to: Stijn Cambie
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #648, https://www.erdosproblems.com/648, accessed 2026-02-13
