On the asymptotic density of the ordered pairs $(a,b)$ of positive integers such that $\gcd (ab,a+b)=\gcd (a,b)$

Consider the arithmetic function of two variables $f(a,b)=\gcd (ab,a+b)\mathrm{/}\gcd (a,b)$, recently investigated by Thang Pang Ern and Malcolm Tan Jun Xi. We deduce asymptotic formulas for sums of the form ${\sum }_{a,b\le x}h(f(a,b))$, where $h$ belongs to a certain class of arithmetic functions. In particular, we obtain an asymptotic formula for the number of ordered pairs $(a,b)\in {\mathbb{N}}^2$ such that $a,b\le x$ and $f(a,b)=m$, where $m\in \mathbb{N}$ is fixed. This shows that in the case $m=1$ the corresponding density is the quadratic class number constant $C={\prod }_p(1-1\mathrm{/}(p^2(p+1)))\doteq 0.881513$. We also formulate some related open problems.