The number of prime factors of integers with dense divisors

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 $C=(1-e^{-\gamma }{)}^{-1}=\mathrm{2.280...}$ and $\gamma$ is Euler's constant. We explore several applications and resolve a conjecture of Margenstern about practical numbers.

arXiv comment18 pages