For an abelian group 𝐺, the Davenport constant $D(G)$ is defined to be the smallest natural number 𝑘 such that any sequence of 𝑘 elements in 𝐺 has a non-empty subsequence whose sum is zero (the identity element). Motivated by some recent developments around the notion of Davenport constant with weights, we study them in some basic cases. We also define a new combinatorial invariant related to $(\mathbb{Z}/n\mathbb{Z})^d$, more in the spirit of some constants considered by Harborth and others and obtain its exact value in the case of $(\mathbb{Z}/n\mathbb{Z})^2$ where 𝑛 is an odd integer.
