A267825 -id:A267825 - cp's OEIS frontend

A267823 Least k such that primorial(n) divides binomial(2k,k).

1, 2, 8, 18, 18, 20, 77, 128, 128, 202, 202, 545, 611, 771, 978, 983, 983, 1625, 2441, 2481, 2481, 2995, 3054, 3284, 3284, 3284, 3284, 3284, 5534, 5534, 5534, 8355, 8355, 10558, 10558, 10558, 45416, 45416, 45416, 45416, 45416, 45416, 45416

Offset: 1

Jonathan Sondow, Jan 20 2016

nonn

The sequence is infinite. In fact, a(n) <= primorial(n)-1 since binomial(2k,k) is divisible by a prime p if and only if some base-p digit of k is >= p/2 (a corollary of Lucas's theorem or Kummer's theorem), and since the 1s digit of primorial(n)-1 in base p is p-1 if p|primorial(n). See the comments in A267825.

What is the explanation for the blocks separated by long gaps: 3284, 3284, 3284, 3284, 3284, then 5534, 5534, 5534, then 8355, 8355, then 10558, 10558, 10558, then 45416, 45416, 45416, 45416, 45416, 45416, 45416?

			C(16,8) is divisible by primorial(3) = 2*3*5 = 30, but C(2k,k) is not divisible by 30 for k < 8, so a(3) = 8.

Chai Wah Wu, Table of n, a(n) for n = 1..150
Wikipedia, Lucas' theorem
Wikipedia, Kummer's theorem

Cf. A000984, A002110, A226078, A267825.

T = Range[100000]; L = {}; n = 1; While[Length[T] > 0,
L = Append[L, First[T]];
T = Select[T, Mod[Binomial[2 #, #], Prime[n + 1]] == 0 &]; n++]; L

a(n) = {my(prn = prod(k=1, n, prime(k)), k = 1); while(binomial(2*k, k) % prn, k++); k;} \\ Michel Marcus, Jan 28 2016

a(n) = min{k : A267825(k) >= n}.

A267825(a(n)) >= n.

cp's OEIS Frontend

A267823 Least k such that primorial(n) divides binomial(2k,k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula