New methods are introduced here to show that if n is a quasiperfect number and u(n) the number of its distinct prime factors, then u(n) > 7 and n > 10 35 , and if further 3}« then u(n) > 9 and n > 1O 40 .

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An argument is outlined which demonstrates that every odd perfect number is divisible by at least eight distinct primes.

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Abstract. Using the circle dissection method, a convergent series and several asymptotic formulae are obtained for p(n, t), the number of partitions of the positive integer n in which no part may be repeated more than t times.