Vlad-Titus Spătaru

Runs of Consecutive Integers Having the Same Number of Divisors

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 $\log\ell_N\ll(\log N\log\log N)^λ$, where $λ=1/2$. Using estimates for the Jacobsthal function, we then improve the result to $λ=1/3$.