On a problem of Erdős concerning primitive sequences

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 \[ \sum \limits _{a \leq n,a \in A} {\frac {1}{{a\log a}} \leq } \sum \limits _{p \leq n,p\;{\text {prime}}} {\frac {1}{{p\log p}}} \quad {\text {for}}\;n > 1.\] This would then partially settle a question of Erdős who asked if this inequality holds for any primitive sequence A.