Suppose, I have a rational number $r$ with $0<r<1$, for example $$r=\frac{53143}{274851}$$
The goal is to write $r$ as a sum of DISTINCT egyptian fractions (fractions with numerator $1$).
The greedy method (always take the smallest possible denominator) leads to the numbers
$$[6, 38, 2706, 34382456, 4763382302120345, 385726786254606395971426367503080]$$
That means $$r=\frac{1}{6}+\frac{1}{38}+\frac{1}{2706}+\frac{1}{34382456}+\frac{1}{4763382302120345}+\frac{1}{385726786254606395971426367503080}$$
The greedy methods usually leads to very large denominators. I would like to have much smaller denominators. I read somewhere that no efficient algorithm is known to find the optimal representation in the sense, that the largest occuring denominator is as small as possible. But I am looking for a method finding a "good" representation in that sense.
A much better solution in the given example would be :
$$[8, 18, 80, 3792, 30539, 48251620]$$
The maximum entry has only $8$ instead of $33$ digits.
Any ideas ?