jmc

Represent the three integers as $n-2$, $n$, and $n+2$, where $n$ is the middle integer. The problem states that $$n(n-2)(n+2)=7(n+(n+2)+(n-2)),$$ which simplifies to $(n-2)(n+2)=21$. Since $7\cdot 3$ and $21\cdot 1$ are the only representations of 21 as a product of two positive integers, we see that $n-2=3$ and $n+2=7$ which implies $n=\boxed{5}$.