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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An upper bound of $ \sum 1/(a\sb i\log a\sb i)$ for primitive sequences


Author: David A. Clark
Journal: Proc. Amer. Math. Soc. 123 (1995), 363-365
MSC: Primary 11B83
MathSciNet review: 1243164
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Abstract: A sequence $ A = \{ {a_i}\} $ of positive integers is called primitive if no term of the sequence divides any other. Erdös conjectures that, for any primitive sequence A,

$\displaystyle \sum\limits_{a \leq n,a \in A} {\frac{1}{{a\log a}} \leq \sum\limits_{p \leq n} {\frac{1}{{p\log p}},\quad {\text{for}}\;n > 1,} } $

where the sum is over all primes less than or equal to n. We show that

$\displaystyle \sum\limits_{a \in A} {\frac{1}{{a\log a}} \leq {e^\gamma } < 1.7811,} $

where $ \gamma $ is Euler's constant.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1243164-0
PII: S 0002-9939(1995)1243164-0
Article copyright: © Copyright 1995 American Mathematical Society