Translated sums on special family of quasi-primitive sequences

A sequence [Formula: see text] of strictly positive integers is said to be quasi-primitive if there are no three distinct terms [Formula: see text] and [Formula: see text] such that [Formula: see text]. Erdős conjectured that the sum [Formula: see text], where [Formula: see text] is the sequence of all powers of prime numbers and [Formula: see text] [Formula: see text]. Recently, Laib et al. proved that the analogous conjecture of Erdős [Formula: see text] is false for [Formula: see text] 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 [Formula: see text].