Weighted real Egyptian numbers
Abstract
Let $\mathcal A = (A_1,\ldots, A_n)$ be a sequence of nonempty finite sets of positive real numbers, and let $\mathcal{B} = (B_1,\ldots, B_n)$ be a sequence of infinite discrete sets of positive real numbers. A weighted real Egyptian number with numerators $\mathcal{A}$ and denominators $\mathcal{B}$ is a real number $c$ that can be represented in the form \[ c = \sum_{i=1}^n \frac{a_i}{b_i} \] with $a_i \in A_i$ and $b_i \in B_i$ for $i \in \{1,\ldots, n\}$. In this paper, classical results of Sierpinski for Egyptian fractions are extended to the set of weighted real Egyptian numbers.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2017
- DOI:
- arXiv:
- arXiv:1708.09478
- Bibcode:
- 2017arXiv170809478N
- Keywords:
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- Mathematics - Number Theory;
- 11D68;
- 11D85;
- 11A67;
- 11B75
- E-Print:
- 10 pages. Improved and corrected