parallel resistors Consider the set $E_b = \left\{1, 1.2, 1.5, 1.8, 2.2, 2.7, 3.3, 3.9, 4.7, 5.6, 6.8, 8.2\right\}$. This is our base set. Let's define the set $E$ as follows:
$$ E = \left\{ 10^k e \mid k=0,1,2,\ldots, \text{for every} e \in E_b \right\}$$
and $\Omega$ as the class of all subsets of $E$. We are intereted in defining a mapping $f$ from $\mathbb{N} \setminus \left\{1 \right\}$ to $\Omega$ such that $f(n)$ is mapped to a set $E_k \in \Omega$ according to the following rule:
$$ \frac{1}{n} = \sum_{e_k \in E_k}{\frac{1}{e_k}}$$
Here are some examples: 


*

* $f(4) = \left\{10, 12, 15\right\}$ as $\frac{1}{4} = \frac{1}{10} + \frac{1}{12} + \frac{1}{15}$

* $f(12) = \left\{12\right\}$

* $f(19) = $ I haven't found a set that could be mapped to $19$, it could be an empty set

* $f(20) = \left\{22, 220 \right\}$ as $\frac{1}{20} = \frac{1}{22} + \frac{1}{220}$


This question/puzzle was brought to my attention by a colleague (an electrical engineer), the idea is to synthesize a desired resistor (an integer-valued in this case) with a given set of standard resistors (set $E$) in parallel (two parallel resistors result in $\frac1{\frac{1}{R_1}+\frac{1}{R_2}}$). 
Now I'm interested to know if we can synthesize a resistor with a minimum numbers of standard resistors, in other words a lower bound on the cardinality of $f(n)$. More interestingly, I'd like to know if someone can come up with an algorithm to construct such a mapping for every $n \in \mathbb{N} \setminus \left\{1 \right\}$, not necessarily with minimum cardinality. Also, for what numbers $f(n)$ is an empty set?
I hope I've articulated the question in mathematical terms clear enough. 

This is a partial answer, identifying some $n$ that cannot be synthesized with a finite subset of values from $E$.
The LCM of $E_{\text{b}}$ is $N=3541509972=2^2\cdot 3^3\cdot 7\cdot 11\cdot 13\cdot 17\cdot 41\cdot 47$. Thus $\frac{N}{e}$ is an integer for $e\in E_{\text{b}}$. For general $e\in E$, $\frac{N}{e}$ therefore has fractional part with finite decimal length.
Consequently, if $f(n)$ is a finite set, then the sum for $\frac{N}{n}$ necessarily has fractional part with finite decimal length.
This implies that if $n$ has prime divisors other than those of $5N$, or prime divisors other than $2$ and $5$ that occur in greater multiplicity than in $N$, no finite set $f(n)$ can be given because $\frac{N}{n}$ would have infinite decimal expansion then. For example $19$ and $81$ cannot be synthesized.
