The mean number of divisors for rough, dense and practical numbers

TL;DR

This paper provides asymptotic estimates for the average number of divisors in special classes of integers, including rough, dense, and practical numbers, confirming a conjecture for practical numbers.

Contribution

It introduces new asymptotic estimates for the mean number of divisors in these classes, notably confirming Margenstern's conjecture for practical numbers.

Findings

Asymptotic estimates for rough and dense numbers' divisors

Confirmation of Margenstern's conjecture for practical numbers

Enhanced understanding of divisor distribution in special integer classes

Abstract

We give asymptotic estimates for the mean number of divisors of integers without small prime factors, integers with bounded ratios of consecutive divisors, and for practical numbers. In the last case, this confirms a conjecture of Margenstern.