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). $$n \quad|\quad R(n) \quad | \quad 2n/\log(n) \quad | \quad (2.30096)n/\log(n)$$ $$10 \quad | \quad 7 \quad | \quad 8.685\ldots \quad | \quad 9.992\ldots$$ $$100 \quad | \quad 44 \quad | \quad 43.429\ldots \quad | \quad 49.964\ldots$$ $$1000 \quad | \quad 339 \quad | \quad 289.529\ldots \quad | \quad 333.098\ldots$$ $$10000 \quad | \quad 2437 \quad | \quad 2171.472\ldots\quad | \quad 2498.235\ldots$$ $$100000 \quad | \quad 18892 \quad | \quad 17371.779\ldots \quad | \quad 19985.884\ldots$$$$1000000 \quad | \quad \text{computing...} \quad | \quad \ldots \quad | \quad \ldots$$
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). $$n \quad|\quad R(n) \quad | \quad 2n/\log(n) \quad | \quad (2.30096)n/\log(n)$$ $$10 \quad | \quad 7 \quad | \quad 8.685\ldots \quad | \quad 9.992\ldots$$ $$100 \quad | \quad 44 \quad | \quad 43.429\ldots \quad | \quad 49.964\ldots$$ $$1000 \quad | \quad 339 \quad | \quad 289.529\ldots \quad | \quad 333.098\ldots$$ $$10000 \quad | \quad 2437 \quad | \quad 2171.472\ldots\quad | \quad 2498.235\ldots$$ $$100000 \quad | \quad 18892 \quad | \quad 17371.779\ldots \quad | \quad 19985.884\ldots$$ $$1000000 \quad | \quad \text{computing...} \quad | \quad \ldots \quad | \quad \ldots$$