A267825 Index of largest primorial factor of binomial(2n,n).

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

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