On a union of homogeneous sequences

A sequence of positive integers is said to be quasi-primitive if there do not exist three distinct integers such that the greatest common divisor of two elements is not the third. It is clear that the sequence of prime numbers is quasi-primitive. The Erdős quasi-primitive conjecture states that the sum of the reciprocal of the product of an element by its logarithm over any quasi-primitive sequence is maximized by the sum over the sequence of all powers of prime numbers. Note that Erdős, before introduce his conjecture, showed that for any quasi-primitive sequence different from one, this sum is convergent and bounded by a real constant. In this article, by studying the analogous Erdős conjecture, we prove that when the real is sufficiently large, there exists a quasi-primitive sequence such that the sum of the reciprocal of the product of an element by the sum of its logarithm and , over this quasi-primitive sequence is very large than the sum over the sequence of all powers of prime numbers. Obviously, if the case where equals zero is also true, then the conjecture is false.