Some new results on consecutive equidivisible integers

TL;DR

This paper investigates the maximum lengths of consecutive integers with a fixed number of divisors, presenting new results including long runs of such integers for specific divisor counts.

Contribution

It provides new findings on the longest runs of consecutive integers with a given number of divisors, including explicit examples for certain divisor counts.

Findings

Longest run of 10 integers with 12 divisors found

Two runs of 14 integers with 24 divisors demonstrated

Maximum possible runs for various fixed divisor counts identified

Abstract

We have found maximum possible runs of consecutive positive integers each having exactly $k$ divisors for some fixed values of $k$. In addition, we exhibit the run of 10 consecutive positive integers each having exactly 12 divisors and two runs of 14 consecutive positive integers each having exactly 24 divisors.