  Let $G$ be a finite abelian group, and let $S$ be a sequence over $G$. Let
  $f(S)$ denote the number of elements in $G$ which can be expressed as the sum
over a nonempty subsequence of $S$. In this paper, we determine all the
sequences $S$ that contains no zero-sum subsequences and $f(S)\leq 2|S|-1$.
