I'm not a math expert so this may be a trivial question; if $p_i$ is the $i$-th prime, let:

$$S(n) = \sum_{i=1}^n p_i$$

be the sum of the first $n$ primes and

$$P(n) = | \{1 \leq i \leq n \mid S(i) \mbox{ is prime} \} | $$

be the number of the sums $S(i), 1 \leq i\ \leq n$ that are prime. Do we have

$$\lim_{n \to \infty} \frac{P(n)}{n} = 0\;?$$

Where can I find a proof?

EDIT: I generated the sequence $P(n)$ and found it on OEIS: Numbers n such that n is prime and is equal to the sum of the first k primes for some k., but there is not a lot of information there.