Egyptian-Fraction Expansions Of REALS

[Marc LeBrun]

 >=Leroy Quet

 > If we have a positive REAL, possibly irrational, x,
 > then we can find a sequence (many many sequences..., I bet) of positive
 > integers {n(k)} such that
 > sum{k=1 to M} 1/n(k) = x,
 > where M is infinity if x is irrational.

NJAStalgia!  Perhaps my earliest correspondence was A006585, which I'd now 
describe as the "number of partitions of 1 into n distinct Egyptian fractions".

It's rewarding to see a comment's been added to this "old" sequence 
recently on 12/30/2003, and an 8th term contributed 1/8/2004 (so the 
"more", and perhaps "hard", keywords should be checked?).

Anyway, take some term e in an Egyptian expansion, and replace it with the 
scaled sum e * U where U is any Egyptian partition of 1 (eg e --> e/2 + e/3 
+ e/6) and you get another Egyptian expansion.

Of course, the reciprocated divisors of perfect numbers gives an infinite 
supply of Egyptian unit partitions.

So there are "very infinitely many" Egyptian expansions of even merely 
rational x (when there are any at all!<;-).

 > ...after dealing with the integer-part differently...

I agree with the sentiment not to dilute the entries for |x|>1 with leading 
1s.  For these I think we should submit the expansion of the fractional 
part and/or reciprocal of x.

 > In any case, I do not believe these sequences are in the Encyclopedia of
 > Integer Sequences, although this idea seems basic to me.
 > I also wonder about alternative EF-expansions, both with all positive
 > terms and with the possibility of having negative terms.

Alas there are EF-relevant sequences in the OEIS that have camouflaged 
presentations (eg lacking the word "Egyptian"!) so finding / 
cross-referencing these is a significant chore.


Regarding expansions generally, it'd be nice to develop some rough 
consistency about which irrational x ought to be covered.  For example 
sqrt(2), the golden ratio, pi and e seem like a basic set.  Euler's gamma 
and zeta(3) are probably close behind.  Perhaps others (eg sqrt(3), 1/e, 
pi^2/6...?)  Any consensus would be helpful as "hints to submitters" (eg 
I've wondered about which "continued surds" I ought to submit).