n + k usually divides C(2n, n) (Pomerance, Theorem 2)

C. Pomerance, Divisors of the middle binomial coefficient, Amer. Math. Monthly 122 (2015) 636–644, Theorem 2: for every positive integer k, the set of n with n + k ∣ C(2n, n) has asymptotic density 1. For primes p ≥ 2k the power of p in n + k always divides C(2n, n) (inequality (4), MiddleBinomial.padicValNat_add_le); for each of the finitely many primes p < 2k, the n where it fails are few, because C(2n, n) has many factors p for most n (Lemma 2 of the paper: few n ≤ x have few carries in base p) and few n ≤ x have p^j ∣ n + k with j large.

Main results

• MiddleBinomial.tendsto_density_add_dvd_centralBinom: for k >= 1 the proportion of n < x with n + k dividing C(2n, n) tends to 1 (Theorem 2)

References

• C. Pomerance, Divisors of the middle binomial coefficient, Amer. Math. Monthly 122 (2015) 636–644, doi:10.4169/amer.math.monthly.122.7.636.

Counting residues in blocks

Large digits

Analysis

Carries

Per-prime bound

theorem MiddleBinomial.tendsto_density_add_dvd_centralBinom (k : ℕ) (hk : 1 ≤ k) : Filter.Tendsto (fun (x : ℕ) => ↑{n ∈ Finset.range x | n + k ∣ n.centralBinom}.card / ↑x) Filter.atTop (nhds 1)

Pomerance, Theorem 2. For every k ≥ 1, the proportion of n < x with n + k ∣ C(2n, n) tends to 1.