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\end{equation}  We know that $H_x$ oscillates around $ln(x)$, and that $H_x \in \mathcal{O}(log(n))$  Thus \[T(n) \in \mathcal{O}(nlog(\sqrt{n})) \Leftrightarrow \Rightarrow  T(n) \in \mathcal{O}(nlog(n))\] \end{document}