# [seqfan] Re: A greedy sequence

franktaw at netscape.net franktaw at netscape.net
Sun Feb 15 00:43:31 CET 2009

```I think you may have misread the reference you gave.  The greedy
algorithm
for unit fractions always terminates (for a positive, rational target).
It is only
the *odd* greedy algorithm (constraining all denominators to be odd)
for
which termination is an unsolved problem.  So there is no question that
all
sub-sequences do, in fact, terminate.

You need to do this kind of computation using exact, "unlimited"
precision
integer arithmetic.  Any floating point arithmetic will soon fail with
this
algorithm.

-----Original Message-----
From: Jeremy Gardiner <jeremy.gardiner at btinternet.com>

The following 'greedy' sequence is formed by summing unit fractions
until
the sum is 1, and repeating using up the 'left over' fractions:

1,
2,3,6,
4,5,7,8,9,10,15,230,57960,
11,12,13,14,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,1544,8242614,
31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,5
5,5
6,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80
,81
,82,202,60402,

(according to my BASIC program implementation: could be errors due to
accuracy of representation)

- anyone else find this interesting, or seen this before?

Likely needs some more investigation before thinking about submitting to
OEIS!

Perhaps related to the The Greedy Algorithm for Unit Fractions:
http://www.mathpages.com/home/kmath454.htm

I guess it may not be known if all of the sub-sequences terminate?

Jeremy

_______________________________________________

Seqfan Mailing list - http://list.seqfan.eu/

```