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