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Problem 779

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Prime Factor and Exponent

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:

$${\displaystyle f_K(n)=\frac{\alpha (n)-1}{(p(n){)}^K}}$$

Also define $\overline{f_K}$ by:

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

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.

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质因数及其指数

对于正整数$n>1$，记$p(n)$为整除$n$的最小质数，并记$\alpha (n)$为$n$的p进数，也即使得$p(n{)}^{\alpha (n)}$整除$n$的最大整数。

对于正整数$K$，定义函数$f_K(n)$为：

$${\displaystyle f_K(n)=\frac{\alpha (n)-1}{(p(n){)}^K}}$$

再定义$\overline{f_K}$为：

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

可以验证，$\overline{f_1}\approx 0.282419756159$。

求${\displaystyle \sum _{K=1}^{\mathrm{\infty }}\overline{f_K}}$，并将你的答案保留$12$位小数。

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