Let $p_{ 1}<p_2<... <p_{ \nu}<...$ 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 $k_{ 1}\le k_{ 2}\le...\le k_{ n}$. Such integers originate in various combinatorial counting problems; when $m=2$, they arise as Matula numbers of certain rooted trees.
Categories: math.NT math.CO