Let $\mathcal P=\{p_1,\dots,p_{2t}\}$ be $2t$ primes between $2^\ell$ and $2^{\ell+1}$ and fix an exponent bound $a\in\mathbb Z_{\geq2}$.
Fix $N\in\mathbb N$ whose prime factors $p$ satisfy $p>2^{\ell+1}$and define $$\mathcal T(\ell,a,\mathcal P)=\big\{\displaystyle\prod_{i=1}^{t}p_{j_i}^{a_{j_i}}:p_{j_i}\neq p_{j_{i'}}\wedge p_{j_i}\in\mathcal P\wedge1\leq j_i\leq2t\wedge1\leq a_{j_i}\leq a\big\}$$
$$\mathcal L(N,\ell,a,\mathcal P)=\{(q,q')\in\mathcal T(\ell,a,\mathcal P)\times\mathcal T(\ell,a,\mathcal P): N|(q-q')\wedge q\neq q'\}.$$
What is the cardinality of the set $\mathcal L(N,\ell,a,\mathcal P)$?
