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\end{split}  \end{equation}  We know that $H_x$ oscillates around $ln(x)$, and that $H_x \in \mathcal{O}(log(n))$ \mathcal{O}(log(n))$\\  Thus \[T(n) \in \mathcal{O}(nlog(\sqrt{n})) \mathcal{O}(nlog(\sqrt{n}))\]  We use the logarithm identity $log_b(x^d) = dlog_b(x)  \Leftrightarrow T(n) \in \mathcal{O}(nlog(n))\] \end{document}