On the reciprocal sum of a sum-free sequence |
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Authors: | YongGao Chen |
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Institution: | 14540. School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing, 210023, China
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Abstract: | Let A = {1 ? a 1 < a 2 < …} 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, Erd?s proved that λ < 103. A sum-free sequence must satisfy a n ? (k + 1)(n ? a k ) 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 a 1 = 1, a 2 = 2, and a 3 = 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. |
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