Asymptotic counting of integers with prime factors $p_{r^a{s}^b}$

Let $p_n$ be the $n$th prime, and let $r,s\ge 2$ be fixed multiplicatively independent integers. We count the integers up to $x$ whose prime factors all have the form $p_{r^a{s}^b}$ with integers $a,b\ge 0$. Our asymptotic formula for this count has relative error $o(1)$ and is explicit down to the multiplicative constant. For $(r,s)=(2,3)$ these integers are the prime codes of the ordinals below ${\omega }^{{\omega }^2}$, so the formula settles that case of the counting problem of Vernaeve, Vindas and Weiermann.