Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$

Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$.

420 satisfies the condition since $7<$ $\sqrt[3]{420}<8$ and $420=\operatorname{lcm}\{1,2,3,4,5,6,7\}$

Suppose $n>420$ is an integer such that every positive integer less than $\sqrt[3]{n}$ divides $n .$

Then $\sqrt[3]{n}>7,$ so $420=\operatorname{lcm}(1,2,3,4,5,6,7)$ divides $n$ thus $n \geq 840$ and $\sqrt[3]{n}>9 .$

Thus $2520=\operatorname{lcm}(1,2, \ldots, 9)$ divides $n$ and $\sqrt[3]{n}>13$

now this pattern looks continues,but i am not able to prove that this pattern always continues..

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