Need assistance in finding sets of solutions for this equation

Question

Given some integer $a$ and $n$, I want to find all integers $b$ and $t_i$ that satisfy this equation:

$$ a = b - \frac{b}{t_1} - \frac{b}{t_2} - \cdots - \frac{b}{t_n} $$

I'm a little confused on how to approach this or what tools to use. Using programming to solve this is also okay.

Note: $a>0$ and $n>0$

I've done one case to see if anything pops out:

For $a = 1$ and $n=2$ I have the complete solution set:

$1 = 6 - \frac{6}{3} -\frac{6}{2}$

$1 = 4 - \frac{4}{2} - \frac{4}{4}$

$1 = 3 - \frac{3}{3} - \frac{3}{3}$

Edit: It looks like a solution is given by $a = (a+n) - \frac{a+n}{a+n}_1 - \cdots - \frac{a+n}{a+n}_n $. And then generative recursion can be used to find more solutions. Greedy algorithm for Egyptian fractions seems really useful.

Answer 1

Rewrite $a = b - \frac{b}{t_1} - \frac{b}{t_2} - \cdots - \frac{b}{t_n} $ as $1-\frac{a}{b} =\sum_{k=1}^n \frac1{t_k} $.

Then look up "Egyptian fractions".

Two sources are

https://en.wikipedia.org/wiki/Egyptian_fraction

and

http://mathworld.wolfram.com/EgyptianFraction.html

Answer 2

I am not sure to be of some help. But since I guess all fractions are irreducible, none of the $t_i$ divide $b$, therefore there product neither, and hence each of them must divide $\Pi_i t_i - \sum_k \Pi_{i\neq k} t_i $ and none of the $\Pi_{i\neq k} t_i$ must equal $b$ nor $\Pi_i t_i$.