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

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.