Differential production cross sections of prompt J/ψand ψ(2S)charmonium and Υ(nS)(n =1, 2, 3) bottomonium states are measured in proton–proton collisions at √s=13 TeV, with data collected by the CMS detector at the LHC, corresponding to an integrated luminosity of 2.3 fb−1for the J/ψand 2.7 fb−1for the other mesons. The five quarkonium states are reconstructed in the dimuon decay channel, for dimuon rapidity |y| <1.2. The double-differential cross sections for each state are measured as a function of yand transverse momentum, and compared to theoretical expectations. In addition, ratios are presented of cross sections for prompt ψ(2S)to J/ψ, Υ(2S)to Υ(1S), and Υ(3S)to Υ(1S)production.