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How to prove that for all positive integers $n$ $$0< \sum_{k=1}^{n}\frac{g(k)}{k}-\frac{2n}{3}<\frac{2}{3}$$ where $g(k)$ denotes the greatest odd divisor of $k$

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    Just a note (not a big deal, since it's easily corrected and I'm not sure whether it's written down in the rules somewhere): I think it's considered bad form to have "display style" (`$$ ... $$` or `\[ ... \]`) math in titles, since it takes up a lot of room on the front page.2012-06-27
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    Have you made any attempt at the problem yourself?2012-06-27

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