Compositional Square

Define a function f(n)=n+n\displaystyle f(n) = n + \left \lfloor \sqrt{n} \right \rfloor.

Find the smallest value of kk such that the composite function,
fk(2017)=ffffk times(2017)f^{k} (2017) = \underbrace{f\circ f \circ f \circ \cdots \circ f}_{k \text{ times}} (2017) can be expressed as m2m^2 for some positive integer mm.

Submit your answer as mkm-k.

Notation: \lfloor \cdot \rfloor denotes the floor function.

This problem is inspired from a Putnam problem.

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