# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## An upper bound of $\sum 1/(a_ i\log a_ i)$ for primitive sequencesHTML articles powered by AMS MathViewer

by David A. Clark
Proc. Amer. Math. Soc. 123 (1995), 363-365 Request permission

## 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, $\sum \limits _{a \leq n,a \in A} {\frac {1}{{a\log a}} \leq \sum \limits _{p \leq n} {\frac {1}{{p\log p}},\quad {\text {for}}\;n > 1,} }$ where the sum is over all primes less than or equal to n. We show that $\sum \limits _{a \in A} {\frac {1}{{a\log a}} \leq {e^\gamma } < 1.7811,}$ where $\gamma$ is Euler’s constant.
References
P. Erdös and Z. Zhang, Upper bound of $\sum {1/({a_i}\log {a_i})}$ for primitive sequences, Proc. Amer. Math. Soc. 117 (1993), 891-895.
• J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689
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