Maximum gap between integers in $[n,n^k]$ having a divisor in $(n,2n)$ (Erdős #693)

Fix $k\ge 2$ and a large integer $n$. Let $A=\{a_1<a_2<\cdots\}$ be the set of integers in $[n,n^k]$ that have at least one divisor in the open interval $(n,2n)$. Erdős asked for an estimate of the maximal gap $G(n,k)=\max_i(a_{i+1}-a_i)$, and specifically whether $G(n,k)\le(\log n)^{O(1)}$. Membership in $A$ is decidable per integer (does $m$ have a divisor in $(n,2n)$?), so for moderate $n$ the whole set $A\cap[n,n^k]$ and its maximal gap are directly computable. GOAL (computational): for $k=2$ (and $k=3$ where feasible) compute $G(n,k)$ for a range of $n$, and fit its growth against $(\log n)^c$ to test the poly-log hypothesis.

Acceptance. PARTIAL / PRIMARY (finite, machine-checkable): for $k=2$ and a range of $n$ (state the range), the exact maximal gap $G(n,2)=\max_i(a_{i+1}-a_i)$ over $A\cap[n,n^2]$, delivered with (i) a runnable script that marks each integer with a divisor in $(n,2n)$ (e.g. by sieving multiples of each $d\in(n,2n)$) and computes gaps, and (ii) a log-log / $(\log n)$ fit assessing whether $G(n,2)\le(\log n)^{O(1)}$. Reporting a maximal gap that grows faster than any fixed power of $\log n$ would be strong evidence against the hypothesis. FULLY RESOLVES: a proof of (or counterexample to) $G(n,k)\le(\log n)^{O(1)}$. State the $n$-range and $k$.