Almost primes of the form ⌊pc⌋⌊pc⌋

Let Pc=(⌊pc⌋)p∈P with c>1c>1 and c∉Nc∉N, where PP is the set of prime numbers, and ⌊⋅⌋⌊⋅⌋ is the floor function. We show that for every such cc there are infinitely many members of Pc having at most R(c)R(c) prime factors, giving explicit estimates for R(c)R(c) when cc is near one and also when cc is large.