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Denote by ω(n) and Ω(n) the number of distinct prime factors of n and the total number of prime factors of n,respectively.For any positive integer ι,we prove that ∑↑2≤n≤x1/ω（n)=ι↑∑↑κ=0(ι↑∑↑i=κ（-1)^i-κCi^κF^(i-κ)(1)κ！x/（loglogx）^i+1+O(x/(loglogx)^ι+2) ∑↑2≤