Asymptotic distribution of integers with certain prime factorizations

Hans Vernaeve; Jasson Vindas; Andreas Weiermann

doi:10.1016/j.jnt.2013.09.001

paper

Asymptotic distribution of integers with certain prime factorizations

arXiv:1303.2498 · doi:10.1016/j.jnt.2013.09.001

Abstract

Let $p_{1} < p_{2} < ... < p_{ν} < ...$ 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 $k_{1} \leq k_{2} \leq ... \leq k_{n}$ . Such integers originate in various combinatorial counting problems; when $m = 2$ , they arise as Matula numbers of certain rooted trees.

11 pages