Divisibility of the central binomial coefficient (²_nⁿ)

We show that for every fixed l ∈ N, the set of n with nl|(²_nⁿ⁾ has a positive asymptotic density cl and we give an asymptotic formula for cl as l → ∞. We also show that #{n ≼ x, (n, (²_nⁿ⁾) = 1} ∼ cx/ log x for some constant c. We use results about the anatomy of integers and tools from Fourier analysis. One novelty is a method to capture the effect of large prime factors of integers in general sequences.