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Given that $\sum_{k=1}^n \lfloor \sqrt{k} \rfloor = Nn-\frac{N(N-1)(2N+5)}{6}$ where $N=\lfloor \sqrt{n} \rfloor$, for which values of N is the sum $\sum_{k=1}^n \lfloor \sqrt{k} \rfloor$ divisible by N?

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