On the equation boldsymbol{n₁n₂=n₃n₄} restricted to factor closed sets

We study the number of solutions $N(B,F)$ of the diophantine equation $n_1n_2=n_3n_4$, where $1\le n_1\le B$, $1\le n_3\le B$, $n_2, n_4\in F$ and $F\subset [1,B]$ is a factor closed set. We study more particularly the case when $F= \big\{m=p_1^{\e_1}\ldots p_k^{\e_k}, \e_j\in \{0,1\}, 1\le j\le k\big\}$, $p_1,\ldots,p_k$ being distinct prime numbers.