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