Asymptotic distribution of integers with certain prime factorizations

Let $p_1<p_2<\mathrm{.}\mathrm{.}\mathrm{.}<p_{\nu }<\mathrm{.}\mathrm{.}\mathrm{.}$ be the sequence of prime numbers and let $m$ be a positive integer. We give a strong asymptotic formula for the distribution of the set of integers having prime factorizations of the form p_{m^{k_1}}p_{m^{k_{2}}...p_{m^{k_{n}}} with $k_1\le k_2\le \mathrm{.}\mathrm{.}\mathrm{.}\le k_n$. Such integers originate in various combinatorial counting problems; when $m=2$, they arise as Matula numbers of certain rooted trees.