For My Friend Nihar Mahajan (Happy Birthday Man)

Algebra Level 5

\[\large{{ x }_{ n+1 }=\left\lfloor \sqrt [ 3 ]{ { x }_{ n }^{ 3 }-\left\lfloor \sqrt [ 3 ]{ { x }_{ n } } \right\rfloor } \right\rfloor }\]

Let there be a sequence defined by above rule for \(n \ge 1\). For \(n \ge k\) the value of \(a_{n}\) will be \(0\), for some positive integer \(k\). Find the value of \(k\), if \(a_{1}=839475687\).

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