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.

1, 3, 7, 19, 17, 47, 131, 41, 43, 1279, 503, 113, 2917, 569, 198623, 14083, 26693, 201611, 42998951, 41931041, 52635749, 1296973, 169097, 1451, 1304394227, 107, 233, 173, 2062225210273, 719, 191, 31551555041, 6301, 563, 3769, 967, 9539, 5073466546857451, 4542977, 9739

Offset: 1

Author

Zhi-Wei Sun, May 05 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 1 for all n > 1.

(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.

Examples

			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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..168

Crossrefs

Cf. A000040, A001006, A002426, A242169, A242171, A242173.

Programs

Mathematica

T[n_]:=Sum[Binomial[n,2k]*Binomial[2k,k],{k,0,n/2}]
f[n_]:=FactorInteger[T[n]]
p[n_]:=Table[Part[Part[f[n],k],1],{k,1,Length[f[n]]}]
Do[If[T[n]<2,Goto[cc]];Do[Do[If[Mod[T[i],Part[p[n],k]]==0,Goto[aa]],{i,1,n-1}];
Print[n," ",Part[p[n],k]];Goto[bb];Label[aa];Continue,{k,1,Length[p[n]]}];
Label[cc];Print[n," ",1];Label[bb];Continue,{n,1,40}]