Properties of nuclei 13,15N and 9B are investigated in the relativistic mean-field theory with NLZ and NL3 force parameters. The calculated binding energies are very close to the experimental ones. The calculations show that the first excited state (1p1/2) in 9B, the first excited state (2s1/2) in 13N and the second excited state (2s1/2) in 15N are weakly bound. In particular, for 13N and 15N, the proton density distributions in the two above excited states have a long tail and the rms radii of the last proton are greatly larger compared with their respective matter radii. It is predicted that a proton halo exists in the first excited state of 13N and in the second excited state of 15N, respectively. It also indicates that the first excited state in 9B is a proton skin state.