On the numbers 𝑛 relatively prime to Ω(𝑛)-𝜔(𝑛)

Abstract

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