Runs of Consecutive Integers Having the Same Number of Divisors

TL;DR

This paper establishes upper bounds on the length of consecutive integer runs with identical divisor counts, improving previous estimates through elementary methods and Jacobsthal function estimates.

Contribution

It provides new upper bounds for the longest runs of consecutive integers with the same number of divisors, refining earlier results with elementary and advanced number theory techniques.

Findings

Upper bound:  \, ext{log}\, \, ext{l}_N \, \\ll \, ( ext{log}\, N \, ext{log}\, ext{log}\, N)^{1/2}

Improved bound:  \, ext{log}\, \, ext{l}_N \, \\ll \, ( ext{log}\, N \, ext{log}\, ext{log}\, N)^{1/3}

Elementary proof technique used for initial bound

Abstract

Our objective is to provide an upper bound for the length $\ell_N$ of the longest run of consecutive integers smaller than $N$ which have the same number of divisors. We prove in an elementary way that $\log\ell_N\ll(\log N\log\log N)^\lambda$, where $\lambda=1/2$. Using estimates for the Jacobsthal function, we then improve the result to $\lambda=1/3$.