A $B_ 2$-sequence with larger reciprocal sum

Abstract:

A sequence of positive integers is called a ${B_2}$-sequence if the pairwise differences are all distinct. The Mian-Chowla sequence is the ${B_2}$-sequence obtained by the greedy algorithm. Its reciprocal sum ${S^\ast }$ has been conjectured to be the maximum over all ${B_2}$-sequences. In this paper we give a ${B_2}$-sequence which disproves this conjecture. Our sequence is obtained as follows: the first 14 terms are obtained by the greedy algorithm, the 15th term is 229, from the 16th term on, the greedy algorithm continues. The reciprocal sum of the first 300 terms of our sequence is larger than ${S^\ast }$.