On the Harborth constant of $C_3\oplus C_{3n}$

Abstract

For a finite abelian group $(G,+,0)$ the Harborth constant $\mathsf{g}(G)$ is the smallest integer $k$ such that each squarefree sequence over $G$ of length $k$, equivalently each subset of $G$ of cardinality at least $k$, has a subsequence of length $\exp (G)$ whose sum is $0$. In this paper, it is established that $\mathsf{g}(G)=3n+3$ for prime $n\ne 3$ and $\mathsf{g}(C_3\oplus C_9)=13$.