For natural numbers $m$ and $n$ I'd like to have some properties of divisors of a triple $(m,n,m+n+1)$. Namely, let $$ f(m,n)=\max\{\mathrm{gcd}(m,n),\mathrm{gcd}(m,m+n+1),\mathrm{gcd}(n,m+n+1)\}, $$ $$ g(m,n)=\min\{\mathrm{gcd}(m,n),\mathrm{gcd}(m,m+n+1),\mathrm{gcd}(n,m+n+1)\}. $$ Denote $$ A=\{(m,n)\in \mathbb N^2\,|\, f(m,n)=1\}, $$ $$ B=\{(m,n)\in \mathbb N^2\,|\, g(m,n)>1\}. $$ Is there a way to describe those sets explicitly, say, by a parametrization?
All I managed to get is that in $A$ both $m$ and $n$ should be odd. Computer simulation didn't help me. In $B$ numbers $m$ and $n$ can be odd or even. For $N\sim 10^2-10^4$ the share of $A$ elements in $[1,N]^2$ is about $1/8$. The share of $B$ is much less but is also stable about ${}\sim 1/54$. Could it mean that the elements are described as residues by some modulus?