Asymptotics for primality of sum of three consecutive primes 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."
 A: 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.
A: For what it's worth, I calculated a bunch of values of $R(n)$ and the claimed densities. I also had into consideration Noam Elkies' possible correction to the factor of $2$ (i.e., use $\lambda n/\log(n)$ instead of $2n/\log(n)$, where $\lambda = \prod_l (1+1/(l-1)^3)\cong 2.30096\ldots$; see the comments in GH's answer).
$$ n \quad|\quad R(n) \quad | \quad 2n/\log(n) \quad | \quad (2.30096)n/\log(n)$$
$$ 10 \quad | \quad 7 \quad | \quad 8.685\ldots \quad | \quad 9.992\ldots$$
$$ 100 \quad | \quad 44 \quad | \quad 43.429\ldots \quad | \quad 49.964\ldots$$
$$ 1000 \quad | \quad 339 \quad | \quad 289.529\ldots \quad | \quad 333.098\ldots$$
$$ 10000 \quad | \quad 2437 \quad | \quad 2171.472\ldots\quad | \quad 2498.235\ldots$$
$$ 100000 \quad | \quad 18892 \quad | \quad 17371.779\ldots \quad  | \quad 19985.884\ldots$$
