Number of certain positive integer solutions of n=x+y+z 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.
 A: 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.
A: If $x$, $y$ and $z$ are chosen "randomly" subject to $x$ and $y$ not both even and $x$ and $z$ not both even, the probability that $x+y+z$ is even is $3/5$.  So it's not surprising that you get different results for odd and even $n$.
Similarly, there should be effects depending on whether $n$ is divisible by each prime: given that $x$ is coprime to $y$ and to $z$ the probability that $x+y+z$ is divisible by $p$ should be $\frac{p^2-1}{p^3 - 2 p + 1}$.  
A: The number of solutions is bounded from below by $cn^2$ (and from above too by obvious reasons). Indeed, let's choose odd $x$ and $y$ at random and independently from the segment $[1,2N]$ for $N$ about $n/100$ and put $z=n-x-y$. For any $d$, the probability that both $x$, $y$ are divisible by $d$ is at most $N^{-2}(N/d+2)^2=d^{-2}+4/Nd+4N^{-2}$. Analogously for $x$ and $z$. Summation over all $d=3,5,7,\dots,n$ gives the upper bound for the probability of at least one event of the form $2(1/9+1/25+\dots)+o(1)$. This is less then $0.8$, so with the probability at least $0.2$ both  pairs $(x,y)$ and $(x,z)$ are coprime. 
