4 added data on divisibility and residue distribution; added 14 characters in body

[Update:] With reference to Noam's comments below, here are some data for the relative frequencies of the sum of three consecutive primes being divisible by the first four odd primes. These are averaged over samples of $400,000$ primes beginning at powers of ten, which are given in the first column; note that these refer to the numbers $x$ themselves, not the indices $n$ of the primes.

$$\log_{10}x&3&5&7&11\\\text{limit?}&0.250&0.188&0.139&0.090$$

I also looked at the joint distribution of the residues modulo $3$ for the three primes. There's a significant preference for avoiding repeated residues; for instance, at $x=10^9$, the repeating patterns $1,1,1$ and $2,2,2$ have relative frequencies around $0.095$, the alternating patterns $1,2,1$ and $2,1,2$ have relative frequencies around $0.150$, and the remaining mixed patterns have relative frequencies around $0.128$, which is almost completely explained by $1,1$ and $2,2$ having relative frequencies $0.445$ and $1,2$ and $2,1$ having relative frequencies $0.555$. I'm trying to work out a probabilistic model for these effects.

Post Undeleted by joriki
3 improved conclusion

I verified Noam's calculation of the factor $\lambda=2.30096\ldots$ and Álvaro's computations, extended the latter and calculated the corresponding ratios:

$$\begin{array}{|c|c|c|c|c|} n & R(n) & 2n/\log n & \lambda n / \log n & R(n)\log n/n\\ \hline 10 & 7 & 9 & 10 & 1.61181\\ 100 & 44 & 43 & 50 & 2.02627\\ 1000 & 339 & 290 & 333 & 2.34173\\ 10000 & 2437 & 2171 & 2498 & 2.24456\\ 100000 & 18892 & 17372 & 19986 & 2.17502\\ 1000000 & 157183 & 144765 & 166549 & 2.17156\\ 10000000 & 1346797 & 1240841 & 1427564 & 2.17078\\ 30000000 & 3784831 & 3484987 & 4009410 & 2.17208\\ \end{array}$$

(The values in the third and fourth columns are rounded to the nearest integer, the values in the last column are rounded to 5 digits after the decimal point.)

Interestingly,

I don't think we can deduce anything from the ratio seems to be converging in this form, however, since it shows convergence in the "random" fluctuations but not with respect to some number roughly halfway between the asymptotic approximations made, e.g. dropping a term $2$ and \log\log n$, which at this stage is still comparable to$\lambda$.\log n$; a more detailed analysis will be required to test the independence hypothesis in this case.

Post Deleted by joriki
2 more sensible rounding
1