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

Runs of Consecutive Integers Having the Same Number of Divisors