Translated sums of primitive sets

Abstract

The Erdős primitive set conjecture states that the sum f (A) = a∈A 1 a log a , ranging over any primitive set A of positive integers, is maximized by the set of prime numbers. Recently Laib, Derbal, and Mechik proved that the translated Erdős conjecture for the sum f (A, h) = a∈A 1 a(log a+h) is false starting at h = 81, by comparison with semiprimes. In this note we prove that such falsehood occurs already at h = 1.04 • • • , and show this translate is best possible for semiprimes. We also obtain results for translated sums of k-almost primes with larger k.