A generalization of primitive sets and a conjecture of Erdős

Abstract

A set of integers greater than 1 is primitive if no element divides another. Erdős proved in 1935 that the sum of $1/(n\log n)$ for $n$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained by the set of prime numbers. We answer the Erdős question in the affirmative for 2-primitive sets. Here a set is 2-primitive if no element divides the product of 2 other elements.

13 pages