From: Eric Jablow <ejablow at cox.net>
Subject: [WSFA] Numbers
Date: Tue, 2 Aug 2005 22:18:16 -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:

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 =CF=83(n) as the sum of the
number's divisors:

=CF=83(4) =3D 1 + 2 + 4 =3D 7.
=CF=83(6) =3D 1 + 2 + 3 + 6 =3D 12.
=CF=83(p) =3D p + 1 for any prime p.

=CF=83(p^n) =3D 1 + p + p^2 + =E2=8B=85 =E2=8B=85 =E2=8B=85 + p^n =3D =
[p^(n+1) - 1]/[p - 1],
     for any prime p and any positive integer n.                  (*)
=CF=83(mn) =3D =CF=83(m) =CF=83(n) if m and n are relatively prime.      =
    (**)

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

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

s(n) =3D =CF=83(n) - n.

Then a perfect number is a number n where s(n) =3D n.

s(6) =3D 6.
s(28) =3D 28.
s(496) =3D 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) =3D n    and    s(n) =3D 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