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We shall give an explicit upper bound for the smallest prime factor ofmultiperfect 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 agiven constant. We shall also give similar results for quasiperfect numbers andrelatively prime amicable pairs of opposite parity.

On the divisibility of odd perfect numbers, quasiperfect numbers and amicable numbers by a high power of a prime