Egyptian-Fraction Expansions Of REALS

[Leroy Quet]

>[posted also to sci.math]
>
>An obvious idea I have not seen *much* about before (which means 
>nothing...).
>
>
>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.
>
>For instance, after dealing with the integer-part differently, we can 
>apply the Greedy-Algorithm to x.
>
>x  =  pi,  as an example:
>
>pi = 3 + 1/8 + 1/61 +...
>
>.....

>
>In any case, I do not believe these sequences are in the Encyclopedia of 
>Integer Sequences, although this idea seems basic to me.
>

>...


Gerry Myerson has posted this list of EIS sequences to sci.math:

>
>%N A014015 Alternating Egyptian fraction expansion of e-2.
>%N A006525 Denominator of Egyptian fraction for e-2.
>%N A069139 Egyptian fraction for square root of 1/2.
>%N A069261 Denominators of the Egyptian fraction for Feigenbaum's 
>constant, 4.6692...
>%N A006487 Egyptian fraction for square root of 2.
>%N A006526 Egyptian fraction for 1/e.
>%N A006524 Egyptian fraction for 1/ pi.
>%N A014013 Alternating Egyptian fraction expansion of pi.
>%N A001466 Denominator of Egyptian fraction for pi - 3.
>

Guess this is an old idea.

thanks,
Leroy Quet