  For a weight-set $A\subseteq \mathbb Z_n$, the $A$-weighted zero-sum constant
$C_A(n)$ is defined to be the smallest natural number $k$, such that any
sequence of $k$ elements in $\mathbb Z_n$ has an $A$-weighted zero-sum
subsequence of consecutive terms. A sequence of length $C_A(n)-1$ in $\mathbb
Z_n$ which does not have any $A$-weighted zero-sum subsequence of consecutive
terms will be called a $C$-extremal sequence for $A$.
  Let $\big(\frac{x}{n}\big)$ denote the Jacobi symbol of $x\in\mathbb Z_n$. We
characterize the $C$-extremal sequences for the weight-set $S(n)=\big\{\,x\in
U(n):\big(\frac{x}{n}\big)=1\,\big\}$ and for the weight-set
$L(n;p)=\big\{\,x\in U(n):\big(\frac{x}{n}\big)=\big(\frac{x}{p}\big)\,\big\}$
where $p$ is a prime divisor of $n$. We can define $D$-extremal sequences for
these weight-sets in a way analogous to the definition of $C$-extremal
sequences. We also characterize these sequences.

Number Theory;Combinatorics