  Let $G$ be a finite additive abelian group. For given $k$ a positive integer,
the $k$-Harborth constant $g^k(G)$ is defined to be the smallest positive
integer $t$ such that given a set $S$ of elements of $G$ with size $t$ there
exists a zero-sum subset of size $k$. We find either the exact value of
$g^k(G)$, or lower and upper bounds for this constant for some groups.
