  Let $G$ be a finite abelian group of exponent $n$, written additively, and
let $A$ be a subset of $\mathbb{Z}$. The constant $s_A(G)$ is defined as the
smallest integer $\ell$ such that any sequence over $G$ of length at least
$\ell$ has an $A$-weighted zero-sum of length $n$ and $\eta_A(G)$ defined as
the smallest integer $\ell$ such that any sequence over $G$ of length at least
$\ell$ has an $A$-weighted zero-sum of length at most $n$. Here we prove that,
for $\alpha \geq \beta$, and $A=\left\{x\in\mathbb{N}\; : \; 1 \le a \le
p^{\alpha} \; \mbox{ and }\; \gcd(a, p) = 1\right \}$, we have
$s_{A}(\mathbb{Z}_{p^{\alpha}}\oplus \mathbb{Z}_{p^\beta}) =
\eta_A(\mathbb{Z}_{p^{\alpha}}\oplus \mathbb{Z}_{p^\beta}) + p^{\alpha}-1 =
p^{\alpha} + \alpha +\beta$ and classify all the extremal $A$-weighted zero-sum
free sequences.
