On the numbers $n$ relatively prime to $\Omega (n)-\omega (n)$

Abstract:

Let $\Omega (n)$ and $\omega (n)$ be the number of all prime factors and different prime factors of $n$, respectively. An elegant result of Alladi showed that the probability of $(n,\Omega (n))=1$ or $(n,\omega (n))=1$ is $6/\pi ^2$. In the other aspect, denoting by $d_k$ the density of the numbers $n$ with $\Omega (n)-\omega (n)=k$, then another elegant formula of Rényi states that $d_k\sim c_0/2^k$ as $k$ tends to infinity, where $c_0=(1/4)\prod _{p>2}(1-(p-1)^{-2})^{-1}$. In this paper, we investigate the numbers $n$ which are relatively prime to $\Omega (n)-\omega (n)$. Let $\mathscr {D}$ denote the set of such numbers. It is proved that there exists a positive constant $c_1$ such that $\mathscr {D}(x)=c_1x+O_{\varepsilon }(\sqrt {x}\log ^{1+\varepsilon }x)$ for any $\varepsilon >0$.