Here, we report on a study of the magnetoresistance (MR) of small-diameter individual Bi and Bi_0.83Sb_0.17 nanowires down to 1.5 K and for magnetic fields up to 14 T. Glass-coated single-crystal microwires were fabricated by the Ulitovsky method. The thin nanowires samples, d < 100 nm, that were investigated displayed pronounced h/e and h/2e resistance oscillations (Aharonov-Bohm (AB) oscillations) as a function of magnetic flux. The observation of these periods is consistent with considering Bi and Bi-Sb nanowires as a tube of surface states. The most intriguing is the presence of MR oscillations equidistant in the magnetic field when the magnetic field is perpendicular to the nanowires axis, when the magnetic flux through the nanowire cross section is zero. In 45-nm Bi nanowires, the self-organization of helical edge states of Bi bilayers led to the formation of series-connected stacks of bilayers, each of which had a closed conducting loop in a transverse magnetic field which results in the appearance of AB oscillations. Apparently, a similar interpretation can be applied to Bi_0.83Sb_0.17 nanowires.