Problem Archive

For a positive integer $n \gt 1$, 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: $$\overline{f_K}=\lim_{N \to \infty} \frac{1}{N}\sum_{n=2}^{N} f_K(n).$$

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

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