New proof and generalization of some results on translated sums over k-almost primes

A sequence 𝒜 of strictly positive integers is said to be primitive if none of its terms divides the others, Erdős conjectured that the sum f(𝒜,0)≤f(ℕ 1 ,0), where ℕ 1 is the sequence of prime numbers and f(𝒜,h)=∑ a∈𝒜 1 a(loga+h). In 2019, Laib et al. proved that the analogous conjecture of Erdős f(𝒜,h)≤f(ℕ 1 ,h) is false for h≥81 on a sequence of semiprimes. Recently, Lichtman gave the best lower bound h=1.04⋯ on semiprimes and he obtained other results for translated sums on k-almost primes with 2<k≤20 and when k sufficiently large. In this note, we propose a new proof of the same result on semiprimes, and we generalize the result on k -almost primes for any k≥2.