ON THE EXISTENCE OF e-MULTIPERFECT NUMBERS

Abstract

The sum of such divisors of N is. denoted by o '(N) , and the number of such divisors by T^\N) . By conventions 1 is an exponential divisor of itself, so that a(l) = 1. The functions j(XN) and o^K^) were introduced in [1] and have been studied in [1] and [2], An integer N is said to be e-perfect whenever o^\N) = 2N9 and g-multiperfect when o^\N) = kN for an integer k > 2. In [1] and [2], several examples of e-perfect numbers are given. It is also proved in [2] that all g-perfect and all g-multiperfect numbers are even. Several unsolved problems are listed in [2], and one of them is whether or not there exists an e-multiperfect number. In this paper, we show that if such a number exists, it must indeed be very, very large.

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