A note on the sum of reciprocals

Abstract

For a fixed positive integer $m$ and any partition $m=m_1+m_2+\cdots +m_e$ , there exists a sequence $\{n_i{\}}_{i=1}^k$ of positive integers such that $$m=\frac{1}{n_1}+\frac{1}{n_2}+\cdots +\frac{1}{n_k},$$ with the property that partial sums of the series $\{\frac{1}{n_i}{\}}_{i=1}^k$ can only represent the integers with the form ${\sum }_{i\in I}{m}_i$, where $I\subset \{1,...,e\}$.

This is a very very preliminary draft, which maybe contains some mistakes