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

1,2

Comments

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?

Links

Chai Wah Wu, Table of n, a(n) for n = 1..100

Wikipedia, Lucas' theorem

Wikipedia, Kummer's theorem

Formula

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

A267825(a(n)) >= n.

Example

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.

Mathematica

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

Prog

(PARI) 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

Crossrefs

Cf. A000984, A002110, A226078, A267825.

Sequence in context: A073601 A051248 A228615 * A063664 A094147 A117612

Adjacent sequences: A267820 A267821 A267822 * A267824 A267825 A267826

Keyword

nonn

Author

Jonathan Sondow, Jan 20 2016

Status

approved