From: Eric Jablow <ejablow at cox.net>
Subject: [WSFA] Numbers [Resent to not use Unicode or HTML]
Date: Tue, 2 Aug 2005 23:10:10 -0400
To: WSFA members <WSFAlist at WSFA.org>
Reply-To: WSFA members <WSFAlist at WSFA.org>

On Aug 2, 2005, at 8:06 PM, I wrote:

Sorry--I tried using the traditional Unicode sigma.  it
didn't work.  Carets stand for exponentiation.

Just to make things clear:

On the other hand, some numbers are amicable: 220 and 284,
for example.  They're even in the Bible!

For nonnegative integers n, define the divisor function sigma(n)
as the sum of the number's divisors:

sigma(4) = 1 + 2 + 4 = 7.
sigma(6) = 1 + 2 + 3 + 6 = 12.
sigma(p) = p + 1 for any prime p.

sigma(p^n) = 1 + p + p^2 + ... + p^n = [p^(n+1) - 1]/[p - 1],
     for any prime p and any positive integer n.                  (*)

sigma(mn) = sigma(m) sigma(n)
      if m and n are relatively prime.          (**)

Rules (*) and (**) allow easy computation of the divisor function.

Then, define the restricted divisor function s(n) as the sum of
the number's proper divisors (those other than n itself):

s(n) = sigma(n) - n.

Then a perfect number is a number n where s(n) = n.

s(6) = 6.
s(28) = 28.
s(496) = 496.

If 2^p  - 1 is [a Mersenne] prime, then 2^(p - 1) (2^p - 1)
is perfect, and all even perfect numbers are of that form.
No odd perfect numbers have yet been found.

Next, if two distinct numbers m and n satisfy

s(m) = n    and    s(n) = m,

then they are called amicable.  The smallest amicable pair
are 220 and 284.

In Genesis 32:14, Jacob gives 220 goats to his brother Esau
out of friendship:

     Two hundred she goats, and twenty he goats, two
     hundred ewes, and twenty rams,

Was that deliberate?  Did Jacob know number theory?

And, John Horton Conway has named certain integers
evil and others odious, in Berlekamp, Conway, and Guy,
"Winning Ways".

The evil numbers are the nonnegative integers with an even
number of 1s in their binary expansion.  The sequence begins
0,3,5,6,9,10,12,15,17,18.  It forms sequence A001969 in Sloane's
on-line encyclopedia.  The complementary sequence forms
the odious numbers, A000069.  These sequences are related to
the nim-values of some of Conway's combinatorial games.

Respectfully,
Eric Jablow