A sequence \(\mathcal {A}\) \({=}\) \((a_{i})_{i\ge 0}\) of strictly positive integers is said to be quasi-primitive if there are no three distinct terms \(a_{i},a_{j}\) and \(a_{k}\in \mathcal {A}\) such that \({(a_{i},a_{j})=a_{k}}\) . Erdős conjectured that the sum \(f(\mathcal {A},0)\le f(Q ,0),\) where \(Q \) is the sequence of all powers of prime numbers and \(f(\mathcal {A},x)=\sum _{a\in \mathcal {A}}\) \(\frac{1}{a\left( \log a+x\right) }\) . It is normal to ask whether the Erdős conjecture remains possible for any \(x\ge 0\) . However, in this observation we show that for x large enough there are an infinite number of primitive sequences such as \( f(\mathcal {A},x)>f(Q ,x)\) . Furthermore, when \(\mathcal {A}\) is a sequence of semiprimes, the last inequality holds for any real number \(x\ge 4.92\) .
