On pairs of consecutive sequences with the same radicals

Let $(m, n, k)$ be a tuple of integers with the property that if i ≤ k, then $m + i$ and $n + i$ have the same radical. Using a result on the abc Conjecture, we bound k from above, improving a result of Balasubramanian, Shorey, and Waldschmidt. We also bound the number of pairs $(m, n)$ for which m < n ≤ x and m(m + 1) ⋯ (m + k - 1)) and n(n + 1) ⋯ (n + - 1) have the same radical and the number of pairs for which $m + i$ and $n + i$ have the same radical for all $i < k$.