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On a problem of Erdős concerning primitive sequences

Author: Zhen Xiang Zhang
Journal: Math. Comp. 60 (1993), 827-834
MSC: Primary 11Y55; Secondary 11B13, 11B83
MathSciNet review: 1181335
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Abstract: A sequence $ A = \{ {a_i}\} $ of positive integers $ {a_1} < {a_2} < \cdots $ is said to be primitive if no term of A divides any other. Let $ \Omega (a)$ denote the number of prime factors of a counted with multiplicity. Let $ p(a)$ denote the least prime factor of a and $ A(p)$ denote the set of $ a \in A$ with $ p(a) = p$. The set $ A(p)$ is called homogeneous if there is some integer $ {s_p}$ such that either $ A(p) = \emptyset $ or $ \Omega (a) = {s_p}$ for all $ a \in A(p)$. Clearly, if $ A(p)$ is homogeneous, then $ A(p)$ is primitive. The main result of this paper is that if A is a positive integer sequence such that $ 1 \notin A$ and each $ A(p)$ is homogeneous, then

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

This would then partially settle a question of Erdős who asked if this inequality holds for any primitive sequence A.

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

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Keywords: Primitive sequences
Article copyright: © Copyright 1993 American Mathematical Society