On the Asymptotic Density of a GCD-based Map

We show that the symmetry of $$f(a,b)=\frac{\gcd (ab,a+b)}{\gcd (a,b)}$$ stems from an ${\mathrm{SL}}_2\left(\mathbb{Z}\right)$ action on primitive pairs and that all solutions to $f(a,b)=n$ admit a uniform three-parameter description – recovering arithmetic-progression families via the Chinese remainder theorem when $n$ is squarefree. It shows that the density of pairs with $f(a,b)=1$ tends to ${\prod }_p(1-p^{-2}(p+1{)}^{-1})\approx 0.88151$, and that its higher-order analogue $f_r$ has a limiting density $6\mathrm{/}{\pi }^2$ for $r\ge 2$.