Translated sums on special family of quasi-primitive sequences

A sequence A = (a(i))(i >= 0) of strictly positive integers is said to be quasi-primitive if there are no three distinct terms a(i),a(j) and a(k) is an element of A such that (ai,aj) = ak. Erdos conjectured that the sum f(A, 0) <= f(Q, 0), where Q is the sequence of all powers of prime numbers and f(A,h) =& sum;(a is an element of A) 1/ a(log a+h). Recently, Laib et al. proved that the analogous conjecture of Erdos f(A,h) <= f(Q,h) is false for h >= 4.92 over the quasi-primitive sequence of semiprimes. In this paper, by constructing a family of quasi-primitive sequences from sequences of k-almost primes and the powers of the prime numbers, we extend this falsity up to 1.46 & ctdot;.