method for representing rational numbers as sums of unit fractions using practical numbers
Fibonacci’s application for practical numbers^{} $n$ was an algorithm to represent proper fractions $\frac{m}{n}$ (with $m>1$) as sums of unit fractions^{} $\sum \frac{{d}_{i}}{n}$, with the ${d}_{i}$ being divisors^{} of the practical number $n$. (By the way, there are infinitely many practical numbers which are also Fibonacci numbers^{}). The method is:

1.
Reduce the fraction to lowest terms. If the numerator is then 1, we’re done.

2.
Rewrite $m$ as a sum of divisors of $n$.

3.
Make those divisors of $n$ that add up to $m$ into the numerators of fractions with $n$ as denominator.

4.
Reduce those fractions to lowest terms, thus obtaining the representation $\frac{m}{n}}={\displaystyle \sum \frac{{d}_{i}}{n}$.
To illustrate the algorithm, let’s rewrite $\frac{37}{42}$ as a sum of unit fractions. Since 42 is practical, success is guaranteed.
At the first step we can’t reduce this fraction because 37 is a prime number^{}. So we go on to the second step, and represent 37 as 2 + 14 + 21. This gives us the fractions
$$\frac{2}{42}+\frac{14}{42}+\frac{21}{42},$$ 
which we then reduce to lowest terms:
$$\frac{1}{21}+\frac{1}{3}+\frac{1}{2},$$ 
giving us the desired unit fractions.
References
 1 M. R. Heyworth, “More on panarithmic numbers” New Zealand Math. Mag. 17 (1980): 28  34
 2 Giuseppe Melfi, “A survey on practical numbers” Rend. Sem. Mat. Univ. Pol. Torino 53 (1995): 347  359
Title  method for representing rational numbers as sums of unit fractions using practical numbers 

Canonical name  MethodForRepresentingRationalNumbersAsSumsOfUnitFractionsUsingPracticalNumbers 
Date of creation  20130322 18:07:00 
Last modified on  20130322 18:07:00 
Owner  PrimeFan (13766) 
Last modified by  PrimeFan (13766) 
Numerical id  4 
Author  PrimeFan (13766) 
Entry type  Algorithm 
Classification  msc 11A25 