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From: bs@faron.UUCP (Robert D. Silverman)
Newsgroups: net.math
Subject: Perfect Numbers
Message-ID: <413@faron.UUCP>
Date: Mon, 16-Dec-85 09:02:56 EST
Article-I.D.: faron.413
Posted: Mon Dec 16 09:02:56 1985
Date-Received: Wed, 18-Dec-85 04:28:33 EST
Distribution: net
Organization: The MITRE Coporation, Bedford, MA
Lines: 41

Here's the latest scoop on perfect number variations: Most of this can be found
in "Unsolved Problems in Number Theory" by Richard Guy, Springer-Verlag.

There are no known odd perfect numbers. The numerical evidence is strong
that none exist. There are none less than 10^200. It must have at least
9 distinct prime divisors. The largest divisor must be greater than 300000
and the second largest is greater than 1000. It is divisible by a prime power
greater than 10^12. If it is not divisible by 3 then it has at least 11 
distinct prime factors. There are lots of other restrictions that could be
given here.
 
Let s(n) be the sum of divisors function acting on n.
 
if s(n) > 2n then n is said to be 'Abundant', if s(n) < 2n then n is said
to be 'Deficient'. I believe it has been proved that the density of abundant
numbers is 3 times that of the deficient numbers. (but I'm not certain)
 

Numbers for which s(n) = 2n-1 are called 'Almost Perfect' Powers of 2 are
'almost perfect'. (trivial) It is not known if there are any others.
 
Numbers for which s(n) = kn are called 'k-fold perfect'. The largest value
of k for which there is a known example is 8. The number is:( I use a dot for
multiplication because it's easier to read)

2.3^23.5^9.7^12.11^3.13^3.17^2.19^2.23.29^2.31^2.37.41.53.61.67^2.71^2.73.83.
89.103.127.131.149.211.307.331.463.521.683.709.1279.2141.2557.5113.6481.10429.
20857.110563.599479.16148168401
 
Erdos conjectures that k = o(ln ln n). 

Pomerance has shown that sum of the reciprocals of all amicable numbers
converges. Amicable numbers are pairs (n,m) such that s(n) = s(m) = m+n.
In fact if A(x) is the number of such pairs with m < n < x then:
A(x) <= x exp(-(ln x)^1/3).
 
If s(n) = 2n+1 then n is said to be 'Quasi-Perfect'.  They must be odd squares
but there are not any known. If there are any then n > 10^35 and it must have
at least 7 distinct prime factors.
 
Bob Silverman