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?


up to $n=1$ there are $1$ prime values, this is $100.000000$ percent
up to $n=10$ there are $4$ prime values, this is $40.000000$ percent
up to $n=100$ there are $10$ prime values, this is $10.000000$ percent
up to $n=1000$ there are $76$ prime values, this is $7.600000$ percent
