Let's call an Egyptian fraction, denoted by $E$, $1$-regular iff there exists some $a \in \mathbb{N}$ such that it can be expressed as folows: $$E = \frac{1}{a} + \frac{1}{a+1} + \frac{1}{a+2} + \frac{1}{a+3} + \dots $$ We generalize this notion of regularity by proposing that an Egyptian fraction $E$ is $n$-regular iff there is some $a \in \mathbb{N}$ such that:
$$E = \frac{1}{a^{n}} + \frac{1}{(a+1)^{n}} + \frac{1}{(a+2)^{n}} + \frac{1}{(a+3)^{n}} + \dots $$
Questions:
- Has this concept of regularity of Egyptian fractions already been proposed elsewhere in the literature on the subject? If so, could you please point towards some sources?
- Can all real numbers be expressed by a $1$-regular Egyptian fraction? The same question applies to $n$-regular Egyptian fractions, for $n \neq 1$.
- Are there any algorithms that are able to convert a real number into a $1$-regular Egyptian fraction?
- Are there any algorithms that are able to convert a real number into an $n$-regular Egyptian fraction, for $n \neq 1$? If so, for which values of $n$ has this been achieved?
Answer to (2).
If I understand correctly, your sums depend on three parameters: $a$, $n$ and the length of the partial sum. Since these three parameters are natural numbers, there are only countably many such sums. Therefore they can not exhaust the set of positive real numbers.