Alexandrian integer

A positive integer $A=pqr$ with nonzero integers $p,q,r$ satisfying $1/A=1/p+1/q+1/r$. Equivalently, $pq+pr+qr=1$. A useful parametrization is $A=n(n+d)(n+e)$ for positive integers $n,d,e$ with $de=n^2+1$, coming from the signed triple $(n,-n-d,-n-e)$.