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Let G be an additive finite abelian group. For every positive integer ℓ, let $dis{c}_{\ell }\left(G\right)$ be the smallest positive integer t such that each sequence S over G of length |S| ≥ t has a nonempty zero-sum subsequence of length not equal to ℓ. In this paper, we determine $dis{c}_{\ell }\left(G\right)$ for certain finite groups, including cyclic groups, the groups $G=C₂\oplus C_{2m}$ and elementary abelian 2-groups. Following Girard, we define disc(G) as the smallest positive integer t such that every sequence S over G with |S| ≥ t has nonempty zero-sum subsequences...