Egyptian-Fraction Expansions Of REALS

[Leroy Quet]

Marc LeBrun wrote:

>....

>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.




I was contemplating the reciprocal of x sequence today.

For instance, how would the EF-expansion of pi (or pi-3) compare with 
that of 1/pi?


>....

>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).

Generally and ideally, perhaps someday a computer could automatically go 
through the EIS database searching for any constants represented as 
decimal-digits, continued fractions, etc, then submit any interesting 
expansions not already in the EIS.
(Would this stretch the database's memory capabilities?)

But for now, perhaps some constants are better suited for certain 
expansions than others, given their use or history or alternative 
representations.

(For instance, the Egyptian fraction expansion of e-2 might be 
interesting, given that e-2 has a well-known sum-formula.)
(As another example related to continued fractions, including in the EIS 
the CF-expansion:
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?A
num=A053978
makes sense, given how the particular constant is originally generated.)

thanks,
Leroy
Quet