Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.2 9/18/84; site utcsri.UUCP
Path: utzoo!utcsri!gclark
From: gclark@utcsri.UUCP (Graeme Clark)
Newsgroups: net.math
Subject: Re: abnormal perfect numbers
Message-ID: <1741@utcsri.UUCP>
Date: Sat, 7-Dec-85 20:59:37 EST
Article-I.D.: utcsri.1741
Posted: Sat Dec  7 20:59:37 1985
Date-Received: Sat, 7-Dec-85 21:54:50 EST
References: <1283@ihuxi.UUCP>
Reply-To: gclark@utcsri.UUCP (Graeme Clark)
Distribution: net
Organization: CSRI, University of Toronto
Lines: 27
Summary: 

In article <1283@ihuxi.UUCP> trough@ihuxi.UUCP (Chris Scussel) writes:
>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.
>
>	Are there any numbers such that s(n)-1 = n?  (That is,
>	sigma(n)-1 = 2*n).
We were interested in just this question.  We called a number n such
that s(n)=n+k a k-perfect number.  Hence you are asking about 1-perfect
numbers.  We wrote a simple program to examine the numbers from 1 to 100,000
looking for k-perfect numbers for k in the range -100 to 100, and found
some curious results:

    There were no 1-perfect numbers.

    All (-1)-perfect numbers were powers of two (it's clear that all
    powers of two are (-1)-perfect, but I don't know about the converse).
    
    For most values of k there were not very many (on the order of 10)
    k-perfect numbers in the range 1-100,000, but for two particular
    values of k there were (12 and 53 I think) there were lots and lots
    of k-perfect numbers (on the order of 1000).  

Does anybody know anything that would shed some light on these results?

				Graeme Clark
				ihnp4!utcsri!gclark