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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

Table of n, a(n) for n=0..100.

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

Cf. A000720, A000984, A002110, A226078, A267823.

Sequence in context: A090822 A091975 A091976 * A151902 A094839 A341771

Adjacent sequences:  A267822 A267823 A267824 * A267826 A267827 A267828

KEYWORD

nonn

AUTHOR

Jonathan Sondow, Jan 27 2016

STATUS

approved

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Last modified September 24 12:23 EDT 2021. Contains 347642 sequences. (Running on oeis4.)