For a positive integer $n>1$, let $p(n)$ be the smallest prime dividing $n$, and let $\alpha (n)$ be its $p$-adic order, i.e. the largest integer such that $p(n{)}^{\alpha (n)}$ divides $n$.

For a positive integer $K$, define the function $f_K(n)$ by: $$f_K(n)=\frac{\alpha (n)-1}{(p(n){)}^K}.$$

Also define $\overline{f_K}$ by: $$\overline{f_K}=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{N}\sum _{n=2}^N{f}_K(n).$$

It can be verified that $\overline{f_1}\approx 0.282419756159$.

Find ${\displaystyle \sum _{K=1}^{\mathrm{\infty }}\overline{f_K}}$. Give your answer rounded to $12$ digits after the decimal point.