Note on the [formula omitted]th power means of distinct prime divisors of composite positive integers

Abstract

Suppose that h is a positive integer. For an integer n ≥ 2 , define P h ( n ) = ( 1 ω ( n ) ∑ p ∣ n p prime p h ) 1 / h , where ω ( n ) denotes the number of distinct prime divisors of n . Let A h ( x ) be the set of all positive integers n ≤ x with ω ( n ) > 1 such that P h ( n ) is prime and P h ( n ) ∣ n . In this paper, we prove that x exp ( 2 h log x log log x ) ≤ | A h ( x ) | ≤ x exp ( ( 1 / 2 ) log x log log x ) , which generalizes a result of Luca and Pappalardi for h = 1 .