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On a question of Erdős concerning cohesive basic sequences


Authors: D. L. Goldsmith and A. A. Gioia
Journal: Proc. Amer. Math. Soc. 34 (1972), 356-358
MSC: Primary 10C10
DOI: https://doi.org/10.1090/S0002-9939-1972-0294285-5
MathSciNet review: 0294285
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Abstract: For an arbitrary basic sequence $ \mathcal{B}$, set $ V(\mathcal{B}) = \{ \char93 {B_k}\vert k \in {Z^ + }\} $, where $ \char93 {B_k}$ is the number of pairs (a, b) in $ \mathcal{B}$ such that $ ab = k$. It is proved that $ V(\mathcal{B})$ is unbounded if either $ \mathcal{B}$ is cohesive or $ \mathcal{B} \not\subset \mathcal{M}$. The set $ V(\mathcal{B})$ is determined explicitly in these cases.


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

  • [1] T. B. Carroll and A. A. Gioia, On extended linear functions, Notices Amer. Math. Soc. 18 (1971), 799. Abstract #71T-A161.
  • [2] A. A. Gioia and D. L. Goldsmith, Convolutions of arithmetic functions over cohesive basic sequences, Pacific J. Math. 38 (1971), 391–399. MR 0309841
  • [3] Donald L. Goldsmith, On the density of certain cohesive basic sequences, Pacific J. Math. 42 (1972), 323–327. MR 0360510

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0294285-5
Keywords: Basic sequence, cohesive, severance class
Article copyright: © Copyright 1972 American Mathematical Society

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