Consider the generalized harmonic numbers evaluated at $2^{-1}$ $$H_{n,2^{-1}} = 1+ {1 \above 1.5pt \sqrt{2}}+ \ldots+ {1 \above 1.5pt \sqrt{n}}$$ The table below lists some initial values:
$$\begin{array}{nc|cccccc} n &1&2&3&4&5&6&7 \\ \hline H_{n,2^{-1}} & 1.00 & 1.71 &2.28&2.78&3.23&3.64&4.01\\ \end{array}$$
Let $$s =min\{n\text{ }|\text{ }H_{n,2^{-1}} \geq n \}$$ For example the smallest $n$ such that $H_{n,2^{-1}}\ge 1$ is $1$. Similarly the smallest $n$ such that $H_{n,2^{-1}}\ge 2$ is $3$. We have the following sequence for $s$ $$\mathfrak a(s) =1,3,5,7,10,14,18,22,\ldots$$ I am asking if the following claim is true -
$$\mathfrak a(s) = \sum_{n\leq s}\Bigg(\Bigg\lfloor{n+2 \above 1.5pt 4}\Bigg\rfloor+\Bigg\lfloor{n+1 \above 1.5pt 4}\Bigg\rfloor\Bigg)$$
Note that $\mathfrak a(s)$ is the sequence A054040.