  A grand canonical system of hard-core bosons in an optical lattice is
considered. The bosons can occupy randomly $N$ equivalent states at each
lattice site. The limit $N\to\infty$ is solved exactly in terms of a
saddle-point integration, representing a weakly-interacting Bose gas. At T=0
there is only a condensate in the limit $N\to\infty$. Corrections in 1/N
increase the total density of bosons but suppress the condensate. This
indicates a depletion of the condensate due to increasing interaction at finite
values of N.
