Let $x$ be a real number.
A best approximation to $x$ for the denominator bound $d$ is a rational number $\frac{r}{s}$ in reduced form, with $s\le d$, such that any rational number which is closer to $x$ than $\frac{r}{s}$ has a denominator larger than $d$:

$|\frac{p}{q}-x|<|\frac{r}{s}-x|\Rightarrow q>d$

For example, the best approximation to $\sqrt{13}$ for the denominator bound 20 is $\frac{18}{5}$ and the best approximation to $\sqrt{13}$ for the denominator bound 30 is $\frac{101}{28}$.

Find the sum of all denominators of the best approximations to $\sqrt{n}$ for the denominator bound ${10}^{12}$, where $n$ is not a perfect square and $1<n\le 100000$.