  We study the condensate and superfluid fraction of a homogeneous gas of
weakly interacting bosons in three spatial dimensions by adopting a
self-consistent Popov approximation, comparing this approach with other
theoretical schemes. Differently from the superfluid fraction, we find that at
finite temperature the condensate fraction is a non-monotonic function of the
interaction strength, presenting a global maximum at a characteristic value of
the gas parameter, which grows as the temperature increases. This non-monotonic
behavior has not yet been observed, but could be tested with the available
experimental setups of ultracold bosonic atoms confined in a box potential. We
clearly identify the region of parameter space that is of experimental interest
to look for this behavior and provide explicit expressions for the relevant
observables. Finite size effects are also discussed within a semiclassical
approximation.
