Results and conjectures related to a conjecture of Erdos concerning primitive sequences

A strictly increasing sequence, finite or infinite, A of positive integers is said to be primitive if no term of A divides any other. Erdos showed that the series Sigma(a is an element of A) 1/a log a, where A is a primitive sequence different from the finite sequence (1), are all convergent and their sums are bounded above by an absolute constant. Besides, he conjectured that. the upper bound of the preceding sums is reached when A is the sequence of the prime numbers. The purpose of this paper is to study the Erdos conjecture. In the first part of the paper, we give two significant conjectures which are equivalent to that of Erdos and in the second one, we study the series of the form Sigma(a is an element of A) 1/a(log a+x) where x is a fixed non-negative real number and A is a primitive sequence different from the finite sequence (1). In particular, we prove that the analog of Erdos's conjecture for these series does not hold, at least for x >= 363. At the end of the paper, we propose a more general conjecture than that of Erdos, which concerns the preceding series, and we conclude by raising some open questions.