We consider the sequence $R\_n=p\_n+p\_{n+1}+p\_{n+2}$, where $\{p\_i\}$ is the prime number sequence, with $p\_0=2$, $p\_1=3$, $p\_2=5$, etc..


The first few values of $R\_n$ are:
10, 15, 23, 31, 41, 49, 59, 71, 83, 97, 109, 121, 131, 143, 159, 173, 187, 199, 211, 223, 235, 251, 269, 287, 301, 311, 319, 329, 349, 371, 395, 407, 425, 439, 457, ...


Now, we define $R(n)$ to be the number of prime numbers in the set $\{R\_0, R\_1 , \dots , R\_n\}$.


What I have found (without justification) is that $R(n) \approx \frac{2n}{\ln (n)}$.


My lack of programming skills, however, prevents me from checking further numerical examples. I was wondering if anyone here had any ideas as to how to prove this assertion.


As a parting statement, I bring up a quote from Gauss, which I feel describes many conjectures regarding prime numbers:
"I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of."



Asymptotics for primality of sum of three consecutive primes
I believe proving (or disproving) such a statement is beyond current technology. On the other hand, by a crude heuristics the conjecture must be right: $R\_1,\dots,R\_n$ are $n$ odd numbers up to $\sim 3n\ln(n)$ which are rather evenly distributed in size and in residue classes. In this range the density of primes among odd numbers is $\sim 2/\ln(n)$, so $R(n)$ should be $\sim 2n/\ln(n)$.


**EDIT:** As Noam Elkies points out in a comment below, there should be a fine tuning similar to the twin prime constant.


