A beautiful limit involving primes and composites I observed the following limit empirically. Let $p_n$ be the $n$-th prime and $c_n$ be the $n$-th composite number then,  

$$
\lim_{n \to \infty}\frac{1}{n}\sum_{i=1}^{n}\frac{p_n c_n}{p_n c_n + p_i c_i} = \frac{\pi}{4}.
$$

I am looking for a proof.
 A: We have that
$$\lim_{n\to\infty}\,\frac1n\,\sum_{k=1}^n\frac n{n+k}=\lim_{n\to\infty}\,\frac1n\,\sum_{k=1}^n\frac1{1+\frac kn}=\int\limits_0^1\frac{dx}{1+x}=\log 2$$
Your sum is not, of course, the above one, but it ressembles it a little, so I'd expect its sum to be closer to $\,\log 2\,$ than to $\,\pi/4\,$ , in particular since the difference between these two numbers is less than $\,1/100\,$ , yet I cannot tell for sure.
A simple program that can add automatically can make some checkings...
A: It turns out that


*

*The limit is correct, but

*It's not saying anything that's very special to primes and composites.


Note that (inspired by DonAntonio's answer)
$$
\lim_{n\to\infty} \frac1n \sum_{k=1}^n \frac{n^2}{n^2 + k^2} 
= \lim_{n\to\infty} \frac1n \sum_{k=1}^n \frac{1}{1 + \left(\frac{k}{n}\right)^2} 
= \int_{0}^{1} \frac{dx}{1 + x^2} = \frac{\pi}{4}
$$
It just so happens that $p_n$ and $c_n$ are (at a very loose level of approximation) on the order of $n$ each, so that $p_n c_n$ is of the order of $n^2$, and therefore your sum
$$
\frac{1}{n}\sum_{k=1}^{n}\frac{p_n c_n}{p_n c_n + p_k c_k}
\approx \frac{1}{n}\sum_{k=1}^{n}\frac{n^2}{n^2 + k^2},
$$
the approximation turning exact in the limit.
To prove this rigorously, we have from the prime number theorem, that $p_n \sim n\ln n$, or to be precise
$$p_n = n\left(\ln n + \ln\ln n - 1 + O\left(\frac{\ln\ln n}{\ln n}\right)\right) = n\left(\ln n + o(\ln n)\right).$$
Similarly for the $n$th composite number $c_n$, we have 
$$c_n = n\left(1 + \frac1{\ln n} + O\left(\frac{1}{\ln^2 n}\right)\right) = n\left(1 + o(1)\right).$$
So $$p_nc_n = n^2\left( \ln n + o(\ln n) \right).$$
Consider a particular value of $\frac{k}{n}$ (say $\alpha$) so that $k = \alpha n$. Then 
$$
\frac{p_kc_k}{p_nc_n} 
= \frac{k^2 (\ln k + o(\ln k))}{n^2(\ln n + o(\ln n))} 
= \frac{k^2}{n^2}\frac{\ln n + \ln \alpha + o(\ln n)}{\ln n + o(\ln n)}
= \frac{k^2}{n^2}(1 + o(1))
$$
Therefore
$$
\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^{n}\frac{p_n c_n}{p_n c_n + p_k c_k}
= \lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^{n}\frac{1}{1 + \frac{p_k c_k}{p_nc_n}}
= \lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^{n}\frac{1}{1 + \frac{k^2}{n^2}(1 + o(1))}
= \int_{0}^{1} \frac{dx}{1 + x^2} = \frac{\pi}{4}.
$$
