AMO-Bench 2025 Problem 43

Let $f(x),g(x),h(x)$ be functions defined on the set of positive integers. Let $\left\lfloor \tfrac{x}{2}\right\rfloor$ denote the greatest integer less than or equal to $\tfrac{x}{2}$. Define $f(x)$ to be the number of integers in the list $1,2,3,\dots,\left\lfloor \tfrac{x}{2}\right\rfloor$ that are coprime to $x$. If $p_{1},p_{2},\dots,p_{k}$ are distinct primes and $y=p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\cdots p_{k}^{\alpha_{k}}$, then $$g(y)=\max\{ p_{1}^{\alpha_{1}}, p_{2}^{\alpha_{2}},\,\dots, p_{k}^{\alpha_{k}}\,\}.$$ For a positive integer $z$, let $h(z)$ be the least positive integer $m$ such that $z\mid m!$. Find all integers $b>1$ such that $$f(b)=h\!\big(g(b^{2})\big).$$