Let G be a finite abelian group of exponent n, written additively, and let A be a subset of Z. The constant s A (G) is defined as the smallest integer ℓ such that any sequence over G of length at least ℓ has an A-weighted zero-sum of length n and η A (G) defined as the smallest integer ℓ such that any sequence over G of length at least ℓ has an A-weighted zero-sum of length at most n. Here we prove that, for α ≥ β, and A = {x ∈ N : 1 ≤ a ≤ p α and gcd(a, p) = 1}, we have s A (Z p α ⊕Z p β ) = η A (Z p α ⊕ Z p β ) + p α − 1 = p α + α + β and classify all the extremal A-weighted zero-sum free sequences.

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