Davenport constant with weights

For the cyclic group $G=\mathbb{Z}\mathrm{/}n\mathbb{Z}$ and any non-empty $A\in \mathbb{Z}$. We define the Davenport constant of $G$ with weight $A$, denoted by $D_A(n)$, to be the least natural number $k$ such that for any sequence $(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_k)$ with $x_i\in G$, there exists a non-empty subsequence $(x_{j_1},\mathrm{.}\mathrm{.}\mathrm{.},x_{j_l})$ and $a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_l\in A$ such that ${\sum }_{i=1}^l{a}_i{x}_{j_i}=0$. Similarly, we define the constant $E_A(n)$ to be the least $t\in \mathbb{N}$ such that for all sequences $(x_1,>\mathrm{.}\mathrm{.}\mathrm{.},x_t)$ with $x_i\in G$, there exist indices $j_1, …, j_n\in$\mathbb{N}$, 1\leq j_1 <… < j_n\leq t$,and$\vartheta_1, >…, \vartheta_n\in A$with$\sum^{n}{i=1} \vartheta_ix{j_i} = 0$\mathrm{.}Inthepresentpaper,weshowthat$E_A(n)=D_A(n)+n-1$. This solve the problem raised by Adhikari and Rath \cite{ar06}, Adhikari and Chen \cite{ac08}, Thangadurai \cite{th07} and Griffiths \cite{gr08}.