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[en] We report Herschel/SPIRE observations of 100 very luminous, optically selected active galactic nuclei (AGNs) at with (erg s−1) ≥ 46.5 , where is λLλ at 1350 Å. The distribution in is similar to the general distribution of Sloan Digital Sky Survey AGNs in this redshift and luminosity interval. We measured star-formation (SF) luminosity, , and SF rate (SFR) in 34 detected sources by fitting combined SF and torus templates, where the torus emission is based on Wide Field Infrared Survey Explorer observations. We also obtained statistically significant stacks for the undetected sources in two luminosity groups. The sample properties are compared with those of very luminous AGNs at . The main findings are: (1) The mean and the median SFRs of the detected sources are and M⊙ yr−1, respectively. The mean SFR of the undetected sources is 148 M⊙ yr−1. The ratio of SFR to the black hole accretion rate is ≈80 for the detected sources and less than 10 for the undetected sources. Unlike a sample of sources at z ≃ 4.8 that we studied recently, there is no difference in and only a very small difference in between the detected and undetected sources. (2) The redshift distribution of and for the most luminous, redshift 2–7 AGNs are different. Similar to previous studies, the highest are found at z ≈ 3. However, the of such sources peaks at z ≈ 5. Assuming the objects in our sample are hosted by the most massive galaxies at those redshifts, we find that approximately 2/3 of the hosts are already below the main sequence of SF galaxies at z = 2−3.5. (3) The spectral energy distributions (SEDs) of dusty tori at high redshift are similar to the shapes found in low redshift, low luminosity AGNs. Herschel upper limits put strong constraints on the long wavelength shape of the SED, ruling out several earlier suggested torus templates as applicable for this sample. (4) We find no evidence for a luminosity dependence of the torus covering factor in sources with (erg s−1) = 44−47.5. This conclusion is based on the recognition that the estimated in several earlier studies is highly uncertain and non-uniformally treated. The median covering factors over this range are 0.68 for isotropic dust emission and 0.4 for anisotropic emission.