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

Let k and r be fixed positive integers with 1< r < k. A positive integer n is called a(k, r)-integer if n is of the form n= a(k)b, where a, b is an element of N and b is r-free. For n with gcd(n,p) = 1, the smallest positive integer f such that n(f )equivalent to 1 (mod p) is called the exponent of n modulo p. If the exponent of n modulo p is p-1, then n is called a primitive root modulo p. Let 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 Sigma(n <= x )A(k,r)(n)A(k,r )(n + 1). 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).