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

Abstract: Let $p_n$ be the $n$th prime, and let $r,s\ge2$ be fixed multiplicatively independent integers. We count the integers up to $x$ whose prime factors all have the form $p_{r^a s<sup>b}$ with integers $a,b\ge0$. 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 $ω<sup>{ω<sup>2}$, so the formula settles that case of the counting problem of Vernaeve, Vindas and Weiermann.