Convergence of the average of f(P1(n)) forces convergence of the average of f(P2(n)) to the same limit for every bounded f.
Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude,Ark
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Rigidity of Averages over the Two Largest Prime Factors
Convergence of the average of f(P1(n)) forces convergence of the average of f(P2(n)) to the same limit for every bounded f.