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A267823 Least k such that primorial(n) divides binomial(2k,k). 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 12 17:17 EDT 2020. Contains 336439 sequences. (Running on oeis4.)