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, the ratio seems to be converging to some number roughly halfway between $2$ and $\lambda$.