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A267825 Index of largest primorial factor of binomial(2n,n). 1
0, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 3, 3, 3, 3, 5, 5, 6, 3, 3, 3, 3, 2, 2, 1, 1, 5, 1, 1, 2, 4, 4, 2, 1, 1, 4, 1, 1, 5, 5, 5, 4, 4, 4, 4, 4, 3, 2, 2, 2, 5, 5, 2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 3, 3, 3, 3, 6, 6, 6, 7, 5, 5, 5, 1, 1, 5, 1, 1, 6, 6, 6, 6, 1, 1, 6, 1, 1, 7, 7, 7, 3, 3, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
For n > 0, binomial(2n,n) is even, so a(n) >= 1.
Is a(n) unbounded? (The largest value for n <= 100000 is a(45416) = 43.)
From Robert Israel, Jan 28 2016: (Start)
a(n) = A000720(p)-1 where p is the least prime that does not divide A000984(n).
Equivalently, p is the least prime such that the base-p representation of n has all digits < p/2.
a(primorial(k)-1) >= k. In particular the sequence is unbounded. (End)
LINKS
Wikipedia, Lucas' theorem
FORMULA
a(A267823(n)) >= n.
min{k : a(k) >= n} = A267823(n).
EXAMPLE
Binomial(16,8) = 12870 is divisible by primorial(3) = 2*3*5 = 30, but not by prime(4) = 7, so a(8) = 3.
MATHEMATICA
PrimorialFactor[n_] := (k = 0; While[Mod[n, Prime[k + 1]] == 0, k++]; k);
Table[PrimorialFactor[Binomial[2 n, n]], {n, 0, 100}]
PROG
(PARI) pf(n) = {my(k = 0); while (n % prime(k+1) == 0, k++); k; }
a(n) = pf(binomial(2*n, n)); \\ adapted from Mathematica; Michel Marcus, Jan 29 2016
CROSSREFS
Sequence in context: A090822 A091975 A091976 * A151902 A094839 A341771
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Jan 27 2016
STATUS
approved

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Last modified April 13 02:05 EDT 2024. Contains 371639 sequences. (Running on oeis4.)