On the distribution of consecutive (k, r)-integer primitive roots modulo p

Let k and r be fixed positive integers with \(1<r<k.\) $1<r<k.$ A positive integer n is called a (k, r)-integer if n is of the form \(n=a^kb,\) $n=a^{k}b,$ where \(a,b\in {\mathbb{N}}\) $a,b\in \mathbb{N}$ and b is r-free. For n with \(\gcd (n, p) = 1,\) $gcd(n,p)=1,$ the smallest positive integer f such that \(n^{f} \equiv 1\; (\bmod\, p)\) $n^f\equiv 1(modp)$ is called the exponent of n modulo p. If the exponent of n modulo p is \(p-1,\) $p-1,$ then n is called a primitive root modulo p. Let \(A_{k,r}(n)\) $A_{k,r}(n)$ be the characteristic function of the (k, r)-integer primitive roots modulo a prime p. In this paper we derive the summation \(\begin{aligned} \sum _{n\le x}A_{k,r}(n)A_{k,r}(n+1). \end{aligned}\) $$\begin{array}{c}\hfill \sum _{n\le x}{A}_{k,r}(n)A_{k,r}(n+1).\end{array}$$ Our result generalizes the previous work about the distribution of consecutive square-free primitive roots by Liu and Dong (Czech Math J 45:247–255, 2015).