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?

Almost certainly yes, but I doubt very much that this can be proven at the current state of the art.

The first few primes that arise are

$$2, 5, 17, 41, 197, 281, 7699, 8893, 22039, 24133, 25237, 28697, 32353, 37561, 38921, 43201, 44683, 55837, 61027, 66463, 70241, 86453$$

See OEIS sequence A013918.

The sum of the first $n$ primes is on the order of $n^2 \log n$, and heuristically a number of this size has probability on the order of $1/\log n$ of being prime. Since $\sum_n 1/\log n = \infty$, there ought to be infinitely many. But that's not a proof.

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 $m\le n$.