A sequence of functions {f_n(q)}_{n=1}^{\infty} satisfies the functional equation for multiplication of quantum integers if f_{mn}(q) = f_m(q)f_n(q^m) for all positive integers m and n. If the functions f_n(q) have only finitely many zeros and poles, then these zeros and poles must be either 0 or roots of unity. This talk describes the structure of all sequences of rational functions with rational coefficients that satisfy this functional equation: They are quotients of products of quantum integers.