Does anyone know how to estimate (as $n$ tends to infinity) the number of solutions of


$$n=x+y+z$$


where $x,y,z$ are positive integers with $x$ coprime to $y$ and to $z$?


Computer experiments suggest that there are roughly $cn^2$ solutions, where $c>0$ is an absolute constant.



Number of certain positive integer solutions of n=x+y+z
For $n$ prime my heuristics tells me that $c=\frac{1}{2}\prod\_p\left(1-\frac{2}{p^2}\right)$, the product being over all primes. Is this supported by computer experiments? If yes, I will share more details.


