theory and/or algorithm to express a proper fraction ...

[jweinberg]

as the addition of proper fractional terms which denominators are single prime numbers and their powers.

Notice that I am not exactly talking here about aliquot ratios:

https://www.encyclopediaofmath.org/inde ... quot_ratio

Yet, I don't know if those two problems are essentially related (I suspect not).

For instance: 60 = (2^2)(3)(5), and

(29/60) = (1/2) - (3/4) + (1/3) + (2/5)

How do you find the numerators, that is: 1, -3, 1 and 2?

For fractions which denominators have simple prime factorizations you can see what the numerators should be, but I have never found a complete theory about such rewriting of proper fractions which numerator and denominator are coprimes as additions of proper fractions which denominators are the factors belonging to the denominator's prime factorization.