Asymptotic distribution of integers with certain prime factorizations

TL;DR

This paper derives a precise asymptotic formula for the distribution of integers with prime factorizations constrained by powers of a fixed prime index, relevant in combinatorial and tree enumeration problems.

Contribution

It provides a strong asymptotic formula for a specific class of integers characterized by prime factorizations involving powers of prime indices.

Findings

Asymptotic distribution formula derived for integers with prime factorizations of a special form.

Application to combinatorial counting problems and Matula numbers of rooted trees.

Enhanced understanding of the distribution of such integers in number theory.

Abstract

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.