$A$-序列倒数和的一个注记

Note on the Reciprocal Sum of a Sum-Free Sequence

中文关键词:  $A$-序列  倒数和  上界.

英文关键词:sum-free sequence  reciprocal sum  upper estimate.

中文摘要:

      在一个无穷整数序列$\{1\leq a_{1}< a_{2}< \cdots \}$中,若没有哪个$a_{i}$是该序列中异于$a_{i}$的若干不同元素之和, 则称该序列为$A$-序列. 本文给出了一个$A$-序列, 其倒数和为 $\sum \frac{1}{a_{i}} > 2.065436491$, 从而改进了关于$A$-序列倒数和的已有结果.

英文摘要:

      An infinite integer sequence $\{1\leq a_{1}< a_{2}< \cdots \} $ is called $A$-sequence, if no $a_{i}$ is sum of distinct members of the sequence other than $a_{i}$. We give an example for the $A$-sequence, and the reciprocal sum of elements is $\sum \frac{1}{a_{i}} > 2.065436491$, which improves slightly the related upper bounds for the reciprocal sums of sum-free sequences.