Is the sum of the first $n$ primes a prime infinitely many times? Define the sequence $P(n)=\sum_{i=1}^{n}p_i$, where $p_i$ is the $i$-th prime number.
I observed for some small $n$ that sometimes this sum evaluates to a prime number, for example $P(2)=2+3=5$ and $P(4)=2+3+5+7=17$ and $P(6)=2+3+5+7+11+13=41$.
So it is natural to ask:

Is it true that there is a sequence of natural numbers $\{n_i:i \in \mathbb N\}$ such that all numbers in the set $\{P(n_i):i \in \mathbb N\}$ are prime numbers?


This is not an answer, but it intend to show that there seems to be an infinite number of primes of the form 
$\sum_{k=1}^n p_k$.
The x-axis in the diagram is $\,^2\!\log n$ and the y-axis is $\,^2\!\log N_n$, where $N_n$ is the number of primes of the form 
$\sum_{k=1}^m p_k$ where $m\le n$.

