Searches are presented for heavy gauge bosons decaying into a top and a bottom quark in data collected by the CMS experiment at s=13 TeV that correspond to an integrated luminosity of 2.2 and 2.6 fb−1 in the leptonic and hadronic analyses, respectively. Two final states are analyzed, one containing a single electron, or muon, and missing transverse momentum, and the other containing multiple jets and no electrons or muons. No evidence is found for a right-handed W′ boson (W′R) and the combined analyses exclude at 95% confidence level W′R with masses below 2.4 TeV if MW′R≫MνR (mass of the right-handed neutrino), and below 2.6 TeV if MW′R<MνR. The results provide the most stringent limits for right-handed W′ bosons in the top and bottom quark decay channel.[Figure not available: see fulltext.].