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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

A $ B\sb 2$-sequence with larger reciprocal sum


Author: Zhen Xiang Zhang
Journal: Math. Comp. 60 (1993), 835-839
MSC: Primary 11B37; Secondary 11B13, 11Y55
MathSciNet review: 1181334
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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}$.


References [Enhancements On Off] (What's this?)

  • [1] P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Enseign. Math. (2) (1980), 52-53.
  • [2] Richard K. Guy, Unsolved problems in number theory, Unsolved Problems in Intuitive Mathematics, vol. 1, Springer-Verlag, New York-Berlin, 1981. Problem Books in Mathematics. MR 656313 (83k:10002)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1993-1181334-7
PII: S 0025-5718(1993)1181334-7
Keywords: $ {B_2}$-sequences, Mian-Chowla sequence, Levine conjecture, greedy algorithm
Article copyright: © Copyright 1993 American Mathematical Society