Irreducible Polynomial Stuff

Consider the set $S$ of all positive integers $n$ such that $x^3-x+n$ is reducible over the rationals (i.e., it factors into at least two nontrivial rational polynomials). For example, 6 is in $S$ since

$x^3-x+6 = (x+2) \left(x^2-2 x+3\right),$

and $x^3-x+n$ is irreducible for all positive integers $n < 6$ . Compute

$\displaystyle\sum_{n \in S} \frac{1}{n}.$