When the Large Divisors of a Natural Number Are in Arithmetic Progression

TL;DR

This paper extends previous work on divisors in arithmetic progression by analyzing large divisors of natural numbers, providing a classification and an asymptotic count formula for such numbers.

Contribution

It introduces the concept of large divisors in arithmetic progression and derives an asymptotic formula for their distribution up to a bound.

Findings

Classifies numbers with large divisors in arithmetic progression.

Derives an asymptotic count formula:  rac{x\, ext{loglog}\,x}{ ext{log}\,x}.

Extends previous work on divisors in arithmetic progression.

Abstract

Iannucci considered the positive divisors of a natural number $n$ that do not exceed the square root of $n$ and found all numbers whose such divisors are in arithmetic progression. Continuing the work, we define large divisors to be divisors at least $\sqrt{n}$ and find all numbers whose large divisors are in arithmetic progression. The asymptotic formula for the count of these numbers up to a bound $x$ is observed to be $\frac{x\log\log x}{\log x}$.