The purpose of this article is to provide the inversion relationships between the reciprocal sum S（1, 2,…, m） and the alternating sum T（1, 2,…, m） for generalized Lucas numbers which generalizes the Melham＇s re...The purpose of this article is to provide the inversion relationships between the reciprocal sum S（1, 2,…, m） and the alternating sum T（1, 2,…, m） for generalized Lucas numbers which generalizes the Melham＇s results.展开更多
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...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.展开更多
An infinite integer sequence {1 ≤ a1 〈 a2 〈 ... } is called A-sequence, if no ai is sum of distinct members of the sequence other than ai. We give an example for the A-sequence, and the reciprocal sum of element...An infinite integer sequence {1 ≤ a1 〈 a2 〈 ... } is called A-sequence, if no ai is sum of distinct members of the sequence other than ai. We give an example for the A-sequence, and the reciprocal sum of elements is∑1/ai〉 2.065436491, which improves slightly the related upper bounds for the reciprocal sums of sum-free sequences.展开更多
基金Supported by the Natural Science Foundation of the Education Department of Henan Province(2003110009)
文摘The purpose of this article is to provide the inversion relationships between the reciprocal sum S（1, 2,…, m） and the alternating sum T（1, 2,…, m） for generalized Lucas numbers which generalizes the Melham＇s results.
基金supported by National Natural Science Foundation of China(Grant No.11071121)
文摘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.
基金the Natural Science Foundation of the Education Department of Sichuan Province (No.2006C057)
文摘An infinite integer sequence {1 ≤ a1 〈 a2 〈 ... } is called A-sequence, if no ai is sum of distinct members of the sequence other than ai. We give an example for the A-sequence, and the reciprocal sum of elements is∑1/ai〉 2.065436491, which improves slightly the related upper bounds for the reciprocal sums of sum-free sequences.
