<s>[INST] On A Splitting Equation of an Egyptian fraction to Egyptian fractions such that all produced fractions have odd denominators. My question is: Do we have an splitting equation where we can produce fractions with odd denominators?
To split an Egyptian fraction to Egyptian fractions, we can use the splitting equation below:
$\frac{1}{n}= \frac{1}{n+1}+\frac{1}{n(n+1)}$ 
The key limitation of the above equation is the following:
If $n$ is even, then $n+1$ is odd and $n(n+1)$ is even, otherwise $n+1$ is even and $n(n+1)$ is even.
Either way, the splitting equation produces with at least one even Egyptian fraction.
An example of a splitting to Odd Egyptian fraction is given below:
$\frac{1}{3}= \frac{1}{5}+\frac{1}{9}+\frac{1}{45}$
$\frac{1}{5}= \frac{1}{9}+\frac{1}{15}+\frac{1}{45}$
$\frac{1}{7}= \frac{1}{15}+\frac{1}{21}+\frac{1}{35}$
$\frac{1}{7}= \frac{1}{9}+\frac{1}{45}+\frac{1}{105}$
$\frac{1}{9}= \frac{1}{15}+\frac{1}{35}+\frac{1}{63}$
$\frac{1}{11}= \frac{1}{21}+\frac{1}{33}+\frac{1}{77}$
The link below is useful for further details about egyptian fraction: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html#section9.5
 [/INST] Just a thought: 
If you limit the problem to just the examples you gave; that is, can any given odd Egyptian fraction be split into three different odd Egyptian fractions?
The examples you gave have two main algebraic forms:
$$\frac{1}{n}=\frac{1}{b}+\frac{1}{an}+\frac{1}{abn}\tag{1}$$
and
$$\frac{1}{n}=\frac{1}{ab}+\frac{1}{an}+\frac{1}{bn}\tag{2}$$
where $a$, $b$ and $n$ are all odd positive integers.
Then one way of proceeding is to study congruence patterns for the two basic forms (1) and (2), for example for $n=4m+1$ and $n=4m+3$ in the first instance.
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