Transition Temperature of a Uniform Imperfect Bose Gas

We calculate the transition temperature of a uniform dilute Bose gas with repulsive interactions, using a known virial expansion of the equation of state. We find that the transition temperature is higher than that of an ideal gas, with a fractional increase $K_0({\mathrm{na}}^3{)}^{1/6}$, where $n$ is the density, $a$ is the $S$-wave scattering length, and $K_0$ is a constant given in the paper. This disagrees with all existing results, analytical or numerical.