On a family of arithmetic series related to the Möbius function

Let $P^{-}(n)$ denote the smallest prime factor of a natural integer $n>1$. Furthermore let $\mu$ and $\omega$ denote respectively the Möbius function and the number of distinct prime factors function. We show that, given any set $\text{scr}P$ of prime numbers with a natural density, we have ${\sum }_{P^{-}(n)\in \text{scr}P}\mu (n)\omega (n)\mathrm{/}n=0$ and provide a effective estimate for the rate of convergence. This extends a recent result of Alladi and Johnson, who considered the case when $\text{scr}P$ is an arithmetic progression.