TITLE: Conjecture: the sequence of sums of all consecutive primes contains an infinite number of primes
QUESTION [21 upvotes]: Starting from 2, the sequence of sums of all consecutive primes is:
$$\begin{array}{lcl}2 &=& 2\\
    2+3 &=& 5 \\
    2+3+5 &=& 10 \\
    2+3+5+7 &=& 17 \\
    2+3+5+7+11 &=& 28 \\
    &\vdots&
\end{array}
$$
If the $n^\text{th}$ prime is $P_n$, then we can write $S_n=\sum_{i=1}^n P_n$.
I conjecture that the sequence $S_n$ contains an infinite number of primes.
I doubt I'm the first person ever to think this, but I cannot find reference to the idea, nor can I conceive a proof, nor a disproof.  Computationally, it can be verified that
    $$S_{13,932}=998,658,581=P_{50,783,012},$$
and the sequence shows no sign of slowing down.
Can you prove the sequence $S_n$ contains an infinite number of primes?

REPLY [3 votes]: The recent paper "Curious conjectures on the distribution of primes among the sums of the first $2n$ primes" by Romeo Meštrović at https://arxiv.org/abs/1804.04198 states your conjecture, along with certain details like the estimate of the number of primes, in section 3.  This paper's other sections examine the distribution of primes, using heuristic, computational and analytic arguments that they are similar to the distribution of primes among the positive integers.