The number of prime factors of integers with dense divisors

2021/01/27 by Weingartner, Andreas
#11N25 #11N37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2101.11585

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^-γ)^-1 = 2.280... and γ is Euler's constant. We explore several applications and resolve a conjecture of Margenstern about practical numbers.

Related

• An Erdős-Kac theorem for integers with dense divisors
• New upper bounds for the number of divisors function
• The mean number of divisors for rough, dense and practical numbers
• Counting integers with a smooth totient
• On the number of distinct prime factors of nj+a^hk