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For positive integers n, let r(n) = #{(x, y, z) ∈ Z : x + y + z = n}. Let g be a positive integer, and let A mod M be any congruence class containing a squarefree integer. We show that there are infinitely many squarefree positive integers n ≡ A mod M for which g divides r(n). This generalizes a result of Cho.

The representation function for sums of three squares along arithmetic progressions