  We shall give an explicit upper bound for the smallest prime factor of
multiperfect numbers of the form $N=p_1^{\alpha_1}\cdots p_s^{\alpha_s}
q_1^{\beta_1}\cdots q_t^{\beta_t}$ with $\beta_1, \ldots, \beta_t$ bounded by a
given constant. We shall also give similar results for quasiperfect numbers and
relatively prime amicable pairs of opposite parity.
