cp's OEIS Frontend

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

%I A267825 #23 Feb 01 2016 05:23:51
%S A267825 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,
%T A267825 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,
%U A267825 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
%N A267825 Index of largest primorial factor of binomial(2n,n).
%C A267825 For n > 0, binomial(2n,n) is even, so a(n) >= 1.
%C A267825 Is a(n) unbounded? (The largest value for n <= 100000 is a(45416) = 43.)
%C A267825 From _Robert Israel_, Jan 28 2016: (Start)
%C A267825 a(n) = A000720(p)-1 where p is the least prime that does not divide A000984(n).
%C A267825 Equivalently, p is the least prime such that the base-p representation of n has all digits < p/2.
%C A267825 a(primorial(k)-1) >= k. In particular the sequence is unbounded. (End)
%H A267825 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas%27_theorem">Lucas' theorem</a>
%F A267825 a(A267823(n)) >= n.
%F A267825 min{k : a(k) >= n} = A267823(n).
%e A267825 Binomial(16,8) = 12870 is divisible by primorial(3) = 2*3*5 = 30, but not by prime(4) = 7, so a(8) = 3.
%t A267825 PrimorialFactor[n_] := (k = 0; While[Mod[n, Prime[k + 1]] == 0, k++]; k);
%t A267825 Table[PrimorialFactor[Binomial[2 n, n]], {n, 0, 100}]
%o A267825 (PARI) pf(n) = {my(k = 0); while (n % prime(k+1) == 0, k++); k;}
%o A267825 a(n) = pf(binomial(2*n, n)); \\ adapted from Mathematica; _Michel Marcus_, Jan 29 2016
%Y A267825 Cf. A000720, A000984, A002110, A226078, A267823.
%K A267825 nonn
%O A267825 0,3
%A A267825 _Jonathan Sondow_, Jan 27 2016