Let $f(x)$ be a non-constant polynomial with integer coefficients. It is a well-known result that if $f(n)$ is a square for all integers $n$, then $f$ must in fact be the square of a polynomial (see, for example: http://www.mast.queensu.ca/~murty/poly2.pdf).

My question is the following. Suppose that $\deg f = d$, and suppose that for every integer $n$, we have that $f(n)$ is a perfect $k$-th power for some $k > 1$ dividing $d$. Can we conclude that $f$ is a perfect $m$-th power, for some $m > 1$ dividing $d$?