Don't Let My Floors Diverge!

Calculus Level 4

\[\large \displaystyle \sum_{n=1}^{\infty} \frac{\lfloor \sqrt{n} \rfloor \lfloor \sqrt[4]{n} \rfloor \lfloor \sqrt[8]{n} \rfloor \lfloor \sqrt[16]{n} \rfloor \dots \lfloor \sqrt[512]{n} \rfloor }{\lfloor \sqrt[1024]{n} \rfloor ^m }\]

Find the minimum integral value of \(m\) such that the above sum converges.

Details and assumptions:

\(\bullet \lfloor x \rfloor \) denotes the floor function.

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