Math Forum

From: haoyuep@aol.com (Dan Hoey)
Date: Mar 24, 2000 2:55 AM
Subject: Re: unitary (Egyptian) fractions

Hugo van der Sanden wrote:

> Bill Taylor wrote:

> > I'd like to plump for a slightly different ordering, which seems
> > much more natural to me; the Farey ordering.  i.e.  Order coarsely
> > by denominator, then more finely by numerator:  1/2 1/3 2/3 1/4
> > 3/4 1/5 2/5 3/5...

> Surely the Farey series is in terms of increasing size, up to a
> given maximum denominator?  F_5 = [ 0, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5,
> 2/3, 3/4, 4/5, 1 ] as I remember it.

There's also another ordering related to Farey series:
1/2,
1/3, 2/3,
1/4, 2/5, 3/5, 3/4,
1/5, 2/7, 3/8, 3/7, 4/7, 5/8, 5/7, 4/5,
1/6, 2/9, 3/11, 3/10, 4/11, 5/13, 5/12, 4/9, 5/9,
where each term (a+c)/(b+d) is formed from two numerically adjacent
predecessors a/b and c/d (including the ghosts 0/1 and 1/1 at the ends).

Is there a name for this sequence?

Dan Hoey <haoyuep@aol.com>