摘要
Let ,4 = {1 ≤ a1 〈 a2 〈 ...} be a sequence of integers. ,4 is called a sum-free sequence if no ai is the sum of two or more distinct earlier terms. Let A be the supremum of reciprocal sums of sum-free sequences. In 1962, ErdSs proved that A 〈 103. A sum-free sequence must satisfy an ≥ （k ~ 1）（n - ak） for all k, n ≥ 1. A sequence satisfying this inequality is called a x-sequence. In 1977, Levine and O＇Sullivan proved that a x-sequence A with a large reciprocal sum must have al = 1, a2 = 2, and a3 = 4. This can be used to prove that λ 〈 4. In this paper, it is proved that a x-sequence A with a large reciprocal sum must have its initial 16 terms： 1, 2, 4, 6, 9, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, and 50. This together with some new techniques can be used to prove that λ 〈 3.0752. Three conjectures are posed.
Let A = {1≤a1<a2<···} be a sequence of integers.A is called a sum-free sequence if no a i is the sum of two or more distinct earlier terms.Let λ be the supremum of reciprocal sums of sum-free sequences.In 1962,Erdo s proved that λ < 103.A sum-free sequence must satisfy a n(k+1)(n-ak) for all k,n 1.A sequence satisfying this inequality is called a κ-sequence.In 1977,Levine and O'Sullivan proved that a κ-sequence A with a large reciprocal sum must have a1=1,a2=2,and a3=4.This can be used to prove that λ < 4.In this paper,it is proved that a κ-sequence A with a large reciprocal sum must have its initial 16 terms:1,2,4,6,9,12,15,18,21,24,28,32,36,40,45,and 50.This together with some new techniques can be used to prove that λ < 3.0752.Three conjectures are posed.
基金
supported by National Natural Science Foundation of China(Grant No.11071121)
