Harmonic sum inequality

Algebra Level 5

\[\large \displaystyle \log_{2}{(2\sqrt{n})} \leq\sum_{i=1}^n \frac{1}{i} \leq\log_{2}{n+1}\]

If the above inequality is satisfied by some set of integer solutions \(n\) then find the sum of the first, third and fifth solutions.

Note: For example, if the solutions in ascending order are 2, 5, 7, 8, 10, ... then the answer will be \(2 + 7 + 10 = 19\).

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