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A242170 Least prime divisor of T(n) which does not divide any T(k) with k < n, or 1 if such a primitive prime divisor of T(n) does not exist, where T(n) is the n-th central trinomial coefficient given by A002426.

%I A242170 #17 May 06 2014 10:57:02
%S A242170 1,3,7,19,17,47,131,41,43,1279,503,113,2917,569,198623,14083,26693,
%T A242170 201611,42998951,41931041,52635749,1296973,169097,1451,1304394227,107,
%U A242170 233,173,2062225210273,719,191,31551555041,6301,563,3769,967,9539,5073466546857451,4542977,9739
%N A242170 Least prime divisor of T(n) which does not divide any T(k) with k < n, or 1 if such a primitive prime divisor of T(n) does not exist, where T(n) is the n-th central trinomial coefficient given by A002426.
%C A242170 Conjecture: (i) a(n) > 1 for all n > 1.
%C A242170 (ii) For any integer n > 3, the n-th Motzkin number M(n) given by A001006 has a prime divisor which does not divide any M(k) with k < n.
%H A242170 Zhi-Wei Sun, <a href="/A242170/b242170.txt">Table of n, a(n) for n = 1..168</a>
%e A242170 a(11) = 503 since T(11) = 3*17*503 with the prime divisor 503 dividing none of T(1),...,T(10), but 3 divides T(2) = 3 and 17 divides T(5) = 51.
%t A242170 T[n_]:=Sum[Binomial[n,2k]*Binomial[2k,k],{k,0,n/2}]
%t A242170 f[n_]:=FactorInteger[T[n]]
%t A242170 p[n_]:=Table[Part[Part[f[n],k],1],{k,1,Length[f[n]]}]
%t A242170 Do[If[T[n]<2,Goto[cc]];Do[Do[If[Mod[T[i],Part[p[n],k]]==0,Goto[aa]],{i,1,n-1}];
%t A242170 Print[n," ",Part[p[n],k]];Goto[bb];Label[aa];Continue,{k,1,Length[p[n]]}];
%t A242170 Label[cc];Print[n," ",1];Label[bb];Continue,{n,1,40}]
%Y A242170 Cf. A000040, A001006, A002426, A242169, A242171, A242173.
%K A242170 nonn
%O A242170 1,2
%A A242170 _Zhi-Wei Sun_, May 05 2014