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A224416 Least prime p such that the polynomial sum_{k=0}^n C_k*x^{n-k} is irreducible modulo p, where C_k denotes the Catalan number binomial(2k,k)/(k+1).

%I A224416 #22 Apr 07 2013 13:26:27
%S A224416 2,3,2,3,17,7,47,3,53,5,137,109,79,11,37,7,59,13,53,251,251,101,467,
%T A224416 149,79,3,83,61,239,31,79,73,73,373,199,5,337,167,17,683,523,269,37,
%U A224416 163,431,163,163,7,487,7,167,163,197,1549,137,503,139,263,151,283
%N A224416 Least prime p such that the polynomial sum_{k=0}^n C_k*x^{n-k} is irreducible modulo p, where C_k denotes the Catalan number binomial(2k,k)/(k+1).
%C A224416 Conjecture: (i) a(n) does not exceed n^2+n+5 for each n>0, and the Galois group of sum_{k=0}^n C_k*x^{n-k} over the rationals is isomorphic to the symmetric group S_n.
%C A224416 (ii) For any positive integer n, the polynomial sum_{k=0}^n binomial(2k,k)*x^{n-k} is irreducible modulo some prime if and only if n is not of the form 2k(k+1), where k is a positive integer.
%C A224416 (iii) For any positive integer n, the polynomial sum_{k=0}^n T_k*x^{n-k} is irreducible modulo some prime not exceeding n^2+n+5, where T_k referes to the central trinomial coefficient A002426(k) which is the coefficient of x^k in the expansion of (x^2+x+1)^k.
%H A224416 Zhi-Wei Sun, <a href="/A224416/b224416.txt">Table of n, a(n) for n = 1..350</a>
%e A224416 a(10) = 5 since sum_{k=0}^{10} C_k*x^{n-k} irreducible modulo 5 but reducible modulo any of 2 and 3.
%e A224416 Note also that a(11) = 137 coincides with 11^2+11+5.
%t A224416 A[n_,x_]:=A[n,x]=Sum[Binomial[2k,k]/(k+1)*x^(n-k),{k,0,n}]
%t A224416 Do[Do[If[IrreduciblePolynomialQ[A[n,x],Modulus->Prime[k]]==True,Print[n," ",Prime[k]];Goto[aa]],{k,1,PrimePi[n^2+n+5]}];
%t A224416 Print[n," ",counterexample];Label[aa];Continue,{n,1,100}]
%Y A224416 Cf. A000040, A000108, A224480, A224417, A224418, A220072, A223934, A224210, A217785, A217788, A224197, A002426.
%K A224416 nonn
%O A224416 1,1
%A A224416 _Zhi-Wei Sun_, Apr 06 2013