On exponentially coprime integers

Abstract

The integers $n={\prod }_{i=1}^r{p}_i^{a_i}$ and $m={\prod }_{i=1}^r{p}_i^{b_i}$ having the same prime factors are called exponentially coprime if $(a_i,b_i)=1$ for every $1\le i\le r$. We estimate the number of pairs of exponentially coprime integers $n,m\le x$ having the prime factors $p_1,...,p_r$ and show that the asymptotic density of pairs of exponentially coprime integers having $r$ fixed prime divisors is $(\zeta (2){)}^{-r}$.