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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
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
Sequence in context: A073601 A051248 A228615 * A063664 A094147 A117612
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Jan 20 2016
STATUS
approved

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Last modified April 15 05:10 EDT 2024. Contains 371667 sequences. (Running on oeis4.)