What are the subsets $V\subseteq\mathbb{Z}^n$ such that there is an integer combination of vectors in $V$ equal to $(1, 1, 1, \ldots)$? (where $n \in \mathbb{Z}^+$ and $\mathbb{Z}^n$ is the n-ary Cartesian product over $\mathbb{Z}$)
By integer combination I mean a positive integer $p$ along with vectors $v_1,v_2,\ldots,v_p \in V$ and integers $z_1,z_2,\ldots,z_p$ such that $z_1v_1 + z_2v_2 + \cdots + z_pv_p = (1,1,1,\ldots)$ with componentwise multiplication and addition.