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