Note on the hhth power means of distinct prime divisors of composite positive integers

Suppose that hh is a positive integer. For an integer n≥2n≥2, define Ph(n)=(1ω(n)∑p∣np primeph)1/h, where ω(n)ω(n) denotes the number of distinct prime divisors of nn. Let Ah(x)Ah(x) be the set of all positive integers n≤xn≤x with ω(n)>1ω(n)>1 such that Ph(n)Ph(n) is prime and Ph(n)∣nPh(n)∣n. In this paper, we prove that xexp(2hlogxloglogx)≤|Ah(x)|≤xexp((1/2)logxloglogx), which generalizes a result of Luca and Pappalardi for h=1h=1.