How many integers whose total number of prime factors is prime are there below x?

Question

Let $ \Omega(n) $ be the total number of prime factors of a positive integer $ n $. Denote by $\mathbb{P}_{\Omega}(x) $ the number of positive integers $ n $ not exceeding $ x $ such that $ \Omega(n)\in\mathbb{P} $. Is an upper bound for this function known ? If yes, is the fact that all non trivial partitions of a prime contain at least $ 2 $ distinct summands anyhow used in establishing this upper bound ?