Journal of the Australian Mathematical Society (Series A)

Research Article

On odd perfect numbers (II), multiperfect numbers and quasiperfect numbers

Graeme L. Cohena1

a1 School of Mathematical Sciences The New South Wales Institute of Technology P.O. Box 123 Broadway NSW 2007 Australia

Let N be a positive integer. This paper is concerned with obtaining bounds for S1446788700021376_inline1 (p prime), when N is an odd perfect number, a multiperfect number, or a quasiperfect number, under assumptions on the existence of such numbers (where none is known) and whether 3 and 5 are divisors. We argue that our new lower bounds in the case of odd perfect numbers are not likely to be significantly improved further. Triperfect numbers are investigated in some detail, and it is shown that an odd triperfect number must have at least nine distinct prime factors.

1980 Mathematics subject classification (Amer. Math. Soc.): 10 A 20.

(Received April 18 1979)

(Revised September 18 1979)