Let $(s_n)_{n\in\mathbb N}$ be defined as follows: For $n\in\mathbb N$, $s_n:=2+3+5+\cdots+p_n$ is the sum of the first $n$ prime numbers (e.g.: $s_1=2$, $s_2=5$, $s_3=10$, $s_4=17$, $\ldots$).

Let $\pi_\sigma(n):=\#\{s_k\mid k\leq n\text{ and }s_k\text{ is prime}\}$ (e.g. $\pi_\sigma(1)=1$, $\pi_\sigma(2)=2$, $\pi_\sigma(3)=2$, $\pi_\sigma(4)=3$, $\ldots$).

Comparing $\pi_\sigma(n)$ with $\frac{n}{2\ln(n)}$ ($\ln$ may denote the natural logarithm) using computer power I suspect $$\pi_\sigma(n)\sim\frac{n}{2\ln(n)}.$$

The following picture compares the two functions on the range from 1 to 50.000.000. The blue line is $\pi_\sigma(n)$ and the orange line is $\frac n{2\ln(n)}$.

$\pi_\sigma(n)$ and $\frac n{2\ln(n)}$ on the range from 1 to 50.000.000" />

My question is: Is this true? Has there already been research made on this question? Does anyone perhabs has prooven this conjecture already?

Until now, I just came up with heuristic arguments to deduce this formula. But these arguments assume for example prime numbers to be randomly distributed within the natural numbers in such a way, that their stochastic distribution is given by the prime number theorem.