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Accoring to H. Lenstra, Chebotarev's theorem holds both for Dirichlet and for natural density (but he doesn't give a reference in this document). Applying Chebotarev to the extension $H/\mathbb{Q}$ where $H$ is the Hilbert class field of $\mathbb{Q}(\sqrt{-n})$ gives the result you want. (At least for primitive discriminants; for non-primitive discriminants youe need an appropriate generalization of the Hilbert class field).