无和数列的倒数和

On the reciprocal sum of sum-free sequences

设A={a1,a2,...}是一个严格递增的正整数数列,如果每一个an都不能写成它前面一些不同项的和,则称A为无和数列.令ρ(A)=∑∞k=11ak.1962年,Erds证明了,对任意无和数列A,有ρ(A)<103.1977年,Levine和O’Sullivan改进为ρ(A)<3.9998.最近,Chen进一步改进为ρ(A)<3.0752.本文证明了,对于无和数列A={a1,a2,...}(a1<a2<···),当a12时,有ρ(A)<2.526.

Let A = {al,a2,...} be a strictly increasing sequence of positive integers. A is called a sum-free 1 1 sequence if no ai is the sum of two or more distinct earlier terms. The number pρ（A）=∑∞k=1 1/ak is called the reciprocal sum of A. In 1962, Erd6s proved that ρ（A） 〈 103 for any sum-free sequence A. In 1977, Levine and O＇Sullivan improved this considerably to ρ（A） 〈 3.9998. Recently, Chen improved it to p（A） 〈 3.0752. In this note, we prove that, for any sum-free sequence A = {a1,a2,...} with 2 ≤ a1 〈 a2 〈 ..., we have p（A） 〈 2.526.