Significant progress toward a theory of high-temperature superconductivity in cuprates has been achieved via the study of effective one- and three-band Hubbard models. Nevertheless, material-specific predictions, while essential for constructing a comprehensive theory, remain challenging due to the complex relationship between real materials and the parameters of effective models. By combining cluster dynamical mean-field theory and density functional theory in a charge self-consistent manner, here we show that material-specific predictions for high-temperature superconductors from first principles are within reach. To demonstrate the capabilities of our approach, we take on the challenge of explaining the remarkable physics of the multilayer cuprates Ca_(1+n)Cu_nO_2nCl₂ and HgBa₂Ca_(n-1Cu_nO_(2n+2). We shed light on the microscopic origin of many salient features of these n-layer cuprates, in particular, differences between the two families and the n dependence of their superconducting properties. We highlight the existence of a minimal doping (4%) required for superconductivity to appear in one of the planes. We interpret the ultimate drop of T_c for n ≥ 4 to the difficulty to dope inner planes. We also find the coexistence of arcs and pockets observed with photoemission, the charge redistribution between copper and oxygen, and the link to the pseudogap. Our work establishes a framework for comprehensive studies of high-temperature superconducting cuprates.