Numbers with three close factorizations

In this paper, we study numbers $n$ that can be factored in three different ways as $n=A_1{B}_1=A_2{B}_2=A_3{B}_3$ where $A_1$, $A_2$, $A_3$ are close to each other and $B_1$, $B_2$, $B_3$ are close to each other.