An upper bound of \(\sum 1/(a_i\log a_i)\) for quasi-primitive sequences (Q1129493)

A sequence \(A=\{a_i\}\) of positive integers is said to be primitive if no term of the sequence divides another. Primitive sequences were investigated by many authors; a celebrated book of Halberstam and Roth contains some basic results. A sequence \(A= \{a_1< a_2<\cdots\}\) is called quasi-primitive if there are no three distinct integers \(a_i,a_j,a_k\in A\) such that \((a_i,a_j)= a_k\). Let \(f(A)= \sum_{a\in A}1/a\log a\). Erdős and Zhang conjectured that for every \(A\), \(f(A)\leq f(P)\), where \(P\) is the sequence of primes and for every quasi-primitive sequence \(A\), \(f(A)\leq f(Q)\), where \(Q\) is the sequence of prime powers. The author previously proved that for every primitive sequence \(f(A)< 1.7811\). Erdős, Sárkőzy and Szemerédi showed that \(f(A)\) is bounded for quasi-primitive sequences \(A\), but an explicit bound is not known. In the present paper the author proves that for every quasi-primitive sequence \(A\), \(f(A)< 4.2022\). It is derived from the following: Let \(\{a_1< a_2<\cdots\}\) be a sequence of integers. Assume that \(a_i q=a_j\) implies \(p(q)< P(a_i)\), where \(p(x)\) is the least prime divisor of \(x\), \(P(x)\) is the largest one. Then \(f(A)\leq 1.83829\).