The number of prime factors of integers with dense divisors

TL;DR

This paper investigates integers with dense divisors, establishing that their typical number of prime factors grows proportionally to  \, log \, log n, and applies this to practical numbers, confirming a conjecture.

Contribution

It provides a new asymptotic formula for the prime factors of dense-divisor integers and resolves a conjecture related to practical numbers.

Findings

Normal order of prime factors is  \, log \, log n for dense-divisor integers.

Confirmed a conjecture of Margenstern about practical numbers.

Established applications of the main result in number theory.

Abstract

We show that for integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the normal order of the number of prime factors is $C \log \log n$, where $C=(1-e^{-\gamma})^{-1} = 2.280...$ and $\gamma$ is Euler's constant. We explore several applications and resolve a conjecture of Margenstern about practical numbers.