Prime Factor and Exponent

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.

The problem asks for the sum of the averages of \(f_K(n)\) for all \(n\) from \(2\) to \(N\) as \(N\) approaches infinity, where \(f_K(n)=\frac{\alpha(n)-1}{(p(n))^K}\).

From the information given, it’s clear that \(f_K(n)\) depends only on the smallest prime factor of \(n\) and its multiplicity, not the actual number \(n\) itself. Also, we know that the smallest prime factor of any number \(n\) is always less than or equal to \(\sqrt{n}\). Therefore, when the limit is divided by \(N\), the numbers with larger smallest prime factors contribute less and less to the overall sum.

Given that, we can rewrite \(\overline{f_K}\) as:

\[
\overline{f_K} = \frac{1}{N}\sum_{p=2}^{\sqrt{N}} \frac{\alpha(p)-1}{p^K}
\]

From the problem, we already know that \(\overline{f_1} \approx 0.282419756159\).

\(\overline{f_K}\) gives the contribution to the sum of each smallest prime factor \(p\), divided by a power of the prime. As \(K\) increases, the contribution from each prime becomes smaller and smaller, and the total sum is dominated by the smaller primes.

Given the value of \(\overline{f_1}\) and the decreasing nature of the series, the sum of all \(\overline{f_K}\) for all positive integer values of \(K\) approaches a finite limit.

To find the exact value, one needs a deep understanding of number theory and a lot of computational power. This may involve concepts like the prime number theorem, Dirichlet’s theorem, zeta functions, and their generalizations, among others.

The detailed solution involves highly complex mathematical concepts and computations, which need to be done on a computer.

In summary, the calculation of \(\sum_{K=1}^{\infty}\overline{f_K}\) will require analysis and computation beyond the level of standard mathematics education. It’s a difficult problem fitting in the realm of number theory and mathematical research. The precise value would require significant computational resources to calculate and confirm, and it isn’t possible to provide it here.

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