一个有较大倒数和的B_2(i≠j)序列(英文)

A B_2(i≠j) Sequence with Larger Reciprocal Sum

如果所有的两项和ai+aj都不同,就称正整数序列a1<a2<…是一个B2-序列.Mian-Chwla序列是用贪婪算法得到的B2-序列,它的倒数和S*曾被猜测为所有B2-序列倒数和的最大值.根据是否允许i=j,相应有两个问题.在允许i=j时,张振祥证明了S*<2.1596及M>2.1597,从而推翻了这个猜测.本文研究不允许i=j(或简称i≠j)的情形.我们给出一个有较大倒数和的B2(i≠j)序列:它的前9项由贪婪算法得到,第10项是54,从第11项起继续用贪婪算法.我们新序列的前200项倒数和大于Main-Chowla(i≠j)序列的倒数和.

A sequence of positive integers a1 〈 a2 〈… is caleed a B2 -sequence if all the sums ai ＋ aj are different. The Mian-Chowla sequence is the B2 -sequence obtained by the greedy algorithm. Its reciprocal sum S^＊ has been conjectured to be the maximum over all B2 -sequences. There are two problems, according as i = j is permitted or not. In case i = j is permitted, Zhang disproves this by showing that S^＊ 〈 2. 1596 and M 〉 2. 1597.In this paper we consider the case i = j is not permitted （or i ≠j for short）. We give aB2（i ≠ j ） sequence with larger reciprocal sum： the first 9 terms are obtained by the greedy algorithm; the 10th term is 54; from the llth term on, the greedy algorithm continues. The reciprocal sum of the first 200 terms of our new sequence is larger than the sum of the reciprocals of the Mian-Chowla （ i ≠ j ） sequence.