Counting consecutive multiplicatively dependent triples

We obtain a nontrivial bound on the number of triples of positive integers $(a,b,c)\in [1,H{]}^3$, which are multiplicatively dependent but each pair of its elements is not, and so is the triple $(a+1,b+1,c+1)$. The method is based on studying factorisations of some resultants and a recent improvement by G. Binyamini, R. Cluckers and F. Kato (2025) of the classical Bombieri–Pila bound.