# Divisibility of the central binomial coefficient $\binom {2n}{n}$

@article{Ford2019DivisibilityOT, title={Divisibility of the central binomial coefficient \$\binom \{2n\}\{n\}\$}, author={Kevin Ford and Sergei Konyagin}, journal={arXiv: Number Theory}, year={2019} }

We show that for every fixed $\ell\in\mathbb{N}$, the set of $n$ with $n^\ell|\binom{2n}{n}$ has a positive asymptotic density $c_\ell$, and we give an asymptotic formula for $c_\ell$ as $\ell\to \infty$. We also show that $\# \{n\le x, (n,\binom{2n}{n})=1 \} \sim cx/\log x$ for some constant $c$. One novelty is a method to capture the effect of large prime factors of integers in general sequences.

## References

SHOWING 1-10 OF 22 REFERENCES

A Rate Estimate in Billingsleys Theorem for the Size Distribution of Large Prime Factors

- Mathematics
- 2000

Let {Pj(m) : 1 j ω(m)} denote the decreasing sequence of distinct prime factors of a positive integer m. We provide an asymptotic expansion for the distribution function Fn(−→ αk) := νn {m : Pj(m) >…

Central binomial coefficients divisible by or coprime to their indices

- Mathematics
- 2017

Let 𝒜 be the set of all positive integers n such that n divides the central binomial coefficient 2n n. Pomerance proved that the upper density of 𝒜 is at most 1 −log2. We improve this bound to 1…

On the asymptotic distribution of large prime factors

- Mathematics
- 1993

A random integer N, drawn uniformly from the set {1,2,..., n), has a prime factorization of the form N = a1a2...aM where ax ^ a2 > ... ^ aM. We establish the asymptotic distribution, as «-»• oo, of…

Multiplicative Number Theory

- Mathematics
- 1967

From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The…

The On-Line Encyclopedia of Integer Sequences

- Computer ScienceElectron. J. Comb.
- 1994

The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used.

Sieve Methods

- Mathematics
- 2001

Preface Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the…

Ten lectures on the interface between analytic number theory and harmonic analysis

- Mathematics
- 1994

Uniform distribution van der Corput sets Exponential sums I: The methods of Weyl and van der Corput Exponential sums II: Vinogradov's method An introduction to Turan's method Irregularities of…

A rate estimate in Billingsley's theorem for the size distribution of large prime factors ∗

- Mathematics
- 2000

Let {Pj(m ):1 j ω(m)} denote the decreasing sequence of distinct prime factors of a positive integer m. We provide an asymptotic expansion for the distribution function Fn( − →

The two-parameter Poisson–Dirichlet point process

- Mathematics
- 2009

The two-parameter Poisson–Dirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete…

On the Prime Factors of ( 2 „ " )

- Mathematics
- 2010

Several quantitative results are given expressing the fact that ( ) is usually divisible by a high power of the small primes. On the other hand, it is shown that for any two primes p and q, there…