Let A⊆ Z_n be a subset. A sequence S = (x₁, ..., x_k) is said to be an A-weighted zero-sum sequence if there exist a₁, ..., a_k ∈ A such that a₁ x₁ + ... + a_k x_k = 0. We refer to A as a weight-set. The A-weighted Davenport constant D_A is defined to be the smallest natural number k such that every sequence of k elements in Z_n has an A-weighted zero-sum subsequence. The constant C_A is defined to be the smallest natural number k such that every sequence of k elements in Z_n has an A-weighted zero-sum subsequence having consecutive terms.

When n is odd, let S(n) be the set of all units in Z_n whose Jacobi symbol with respect to n is 1. We compute the constants C_S(n) and D_S(n). For a prime divisor p of n, we also compute these constants for a related weight-set L(n;p). This is the set of all units x in Z_n such that the Jacobi symbol of x with respect to n is the same as the Legendre symbol of x with respect to p. We show that even though these weight-sets A may have half the size of U(n) (which is the set of units of Z_n), the corresponding A-weighted constants are the same as those for the weight-set U(n).