Curious conjectures on the distribution of primes among the sums of the first $2n$ primes

Abstract: Let $p_n$ be $n$th prime, and let $(S_n){n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding computational results suggest that the primes are distributed among sequence $(S_n)$ in the same way that they are distributed among positive integers. In other words, taking into account the Prime Number Theorem, this assertion is equivalent to \begin{equation*}\begin{split} &# {p:\, p\,\,{\rm is\,\,a\,\, prime\,\, and}\,\, p=S_k \,\,{\rm for\,\,some\,\,} k \,\,{\rm with\,\,} 1\le k\le n} \sim & # {p:\, p\,\,{\rm is\,\,a\,\, prime\,\, and}\,\, p=k \,\,{\rm for\,\,some\,\,} k \,\,{\rm with\,\,} 1\le k\le n}\sim\frac{\log n}{n}\,\, {\rm as}\,\, n\to\infty, \end{split}\end{equation*} where $|S|$ denotes the cardinality of a set $S$. Under the assumption that this assertion is true (Conjecture 3.3), we say that $(S_n)$ satisfies the Restricted Prime Number Theorem. Motivated by this, in Sections 1 and 2 we give some definitions, results and examples concerning the generalization of the prime counting function $\pi(x)$ to increasing positive integer sequences. The remainder of the paper (Sections 3-7) is devoted to the study of mentioned sequence $(S_n)$. Namely, we propose several conjectures and we prove their consequences concerning the distribution of primes in the sequence $(S_n)$. These conjectures are mainly motivated by the Prime Number Theorem, some heuristic arguments and related computational results. Several consequences of these conjectures are also established.