A082916 Numbers k such that k and binomial(2*k, k) are relatively prime.

0, 1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 55, 59, 61, 67, 71, 73, 79, 81, 83, 89, 93, 97, 101, 103, 107, 109, 111, 113, 119, 121, 125, 127, 131, 137, 139, 149, 151, 155, 157, 161, 163, 167, 169, 173, 179, 181, 185, 191, 193, 197

Offset: 1

Author

Benoit Cloitre, May 25 2003

Keywords

• nonn

Comments

Also the numbers k such that every base-p digit of k is less than p/2, for every prime divisor p of k. Contains all odd primes and their powers. - David Radcliffe, Jun 28 2025

References

• J. Glaisher, On the residue of a binomial-theorem coefficient with respect to a prime modulus, Quarterly J. of Pure and Applied Math. 30 (1899), 150-156.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000984 (central binomial coefficients). Contains A061345.

Programs

Mathematica

Select[Range[0, 100], CoprimeQ[Binomial[2*#, #], #] &] (* Amiram Eldar, May 24 2020 *)

PARI

isok(n) = gcd(n, binomial(2*n, n)) == 1; \\ Michel Marcus, Dec 04 2013

Python

from math import gcd
A082916_list, b = [], 1
for n in range(10**5):
    if gcd(n,b) == 1:
        A082916_list.append(n)
    b = b*(4*n+2)//(n+1) # Chai Wah Wu, Mar 25 2016

Formula

It seems that a(n) is asymptotic to c*n*log(n) with 0.7

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