Runs of Consecutive Integers Having the Same Number of Divisors - Spătaru, Vlad-Titus (The PUMP Journal of Undergraduate Research, 2023)
Our objective is to provide an upper bound for the length ℓN 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 ℓN << (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.