  We consider the Schr\"odinger operator with a periodic potential on a quasi
1D continuous periodic model of armchair nanotubes in $\R^3$ in a uniform
magnetic field (with amplitude $B\in \R$), which is parallel to the axis of the
nanotube. The spectrum of this operator consists of an absolutely continuous
part (spectral bands separated by gaps) plus an infinite number of eigenvalues
with infinite multiplicity. We describe all eigenfunctions with the same
eigenvalue including compactly supported. We describe the spectrum as a
function of $B$. For some specific potentials we prove an existence of gaps
independent on the magnetic field. If $B\ne 0$, then there exists an infinite
number of gaps $G_n$ with the length $|G_n|\to\iy$ as $n\to\iy$, and we
determine the asymptotics of the gaps at high energy for fixed $B$. Moreover,
we determine the asymptotics of the gaps $G_n$ as $B\to 0$ for fixed $n$.
