Let $d(n)$ denote the largest divisor of $n$ less than $\sqrt{n}$. Are there good lower bounds for $d$ that hold for almost all natural numbers?

More precisely, is there a function $f$, say $f(n)=\frac{\sqrt{n}}{(\log{n})^{100}}$, such that for large $x$, $d(n)\geq f(n)$ holds for almost all $n\leq x$.

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