For all primes $p>2$, does the following identity
$$\frac{1-\zeta(1-p,p)}{p}\in\mathbb{Z}$$
hold such that $\zeta(s,a)$ denotes the Hurwitz zeta function? If so, how would I go about formulating a proof (or is there a proof already in the literature)? Any help would be much appreciated.
The first few values are
