You might be interested in a paper of Filip Najman at arxiv.org/abs/1108.3710 . The largest prime divisor of the product n+1 to n +f(k) is looked at and shown to be larger than k when n is larger than k. f(k) is pretty small and conjectured to be O(log(k)^2). So you don't need as many as k consecutive integers. Gerhard "Sees This As Smooth Intervals" Paseman, 2015.11.02