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From: trough@ihuxi.UUCP (Chris Scussel)
Newsgroups: net.math
Subject: abnormal perfect numbers
Message-ID: <1283@ihuxi.UUCP>
Date: Thu, 5-Dec-85 09:08:40 EST
Article-I.D.: ihuxi.1283
Posted: Thu Dec  5 09:08:40 1985
Date-Received: Fri, 6-Dec-85 07:11:45 EST
Distribution: net
Organization: AT&T Bell Laboratories
Lines: 21

A number is perfect if sigma(n) = 2*n. An alternate definition is when
s(n) = n, where s(n) is the sum of the divisors of the number, including 1
but not including n. Let's consider this definition "unfair"; 1 should be
excluded too, because 

	1) if k is a factor of n then n/k should be too
	2) if k is a factor of n then k itself is the product of primes
           unless k is 1

Even if the above reasons aren't very convincing, they lead to an interesting
question:


	Are there any numbers such that s(n)-1 = n?  (That is,
	sigma(n)-1 = 2*n).

This is similar to another posted problem: s(x)-x = n, where n=-1.

				Chris Scussel
				AT&T Bell Labs
				ihnp4!ihuxi!trough