  Let $g(n)$ be the largest positive integer $k$ such that there are distinct
primes $p_i$ for $1\leq i\leq k$ so that $p_i |n+i$. This function is related
to a celebrated conjecture of C.A. Grimm. We establish upper and lower bounds
for $g(n)$ by relating its study to the distribution of smooth numbers.
Standard conjectures concerning smooth numbers in short intervals imply $g(n)
=O(n^\epsilon)$ for any $\epsilon >0$. We also prove unconditionally that $g(n)
=O(n^\al)$ with $0.45<\al <0.46$. The study of $g(n)$ and cognate functions has
some interesting implications for gaps between consecutive primes.
