cp's OEIS Frontend

A236308 Number of primes q < prime(n)/2 such that the Catalan number C(q) is a primitive root modulo prime(n).

%I A236308 #21 Aug 05 2019 02:48:38
%S A236308 0,0,1,1,1,1,1,1,4,2,1,4,1,3,3,5,5,5,2,4,5,4,10,4,7,7,4,7,4,9,5,6,10,
%T A236308 9,7,5,5,12,12,13,12,4,10,7,13,4,7,10,18,9,14,13,9,9,15,17,16,8,9,12,
%U A236308 10,19,13,10,14,14,13,6,18,18,14,24,13,16,9,22,20,12,23,15
%N A236308 Number of primes q < prime(n)/2 such that the Catalan number C(q) is a primitive root modulo prime(n).
%C A236308 Conjecture: a(n) > 0 for all n > 2. In other words, for any prime p > 3, there exists a prime q < p/2 such that the Catalan number C(q) = binomial(2q, q)/(q+1) is a primitive root modulo p.
%C A236308 We have verified this for all n = 3, ..., 2*10^5.
%H A236308 Zhi-Wei Sun, <a href="/A236308/b236308.txt">Table of n, a(n) for n = 1..1000</a>
%H A236308 Z.-W. Sun, <a href="http://arxiv.org/abs/1405.0290">New observations on primitive roots modulo primes</a>, arXiv preprint arXiv:1405.0290 [math.NT], 2014.
%e A236308 a(13) = 1 since C(7) = 429 is a primitive root modulo prime(13) = 41.
%t A236308 f[k_]:=CatalanNumber[Prime[k]]
%t A236308 dv[n_]:=Divisors[n]
%t A236308 Do[m=0;Do[If[Mod[f[k],Prime[n]]==0,Goto[aa],Do[If[Mod[f[k]^(Part[dv[Prime[n]-1],i]),Prime[n]]==1,Goto[aa]],{i,1,Length[dv[Prime[n]-1]]-1}]];m=m+1;Label[aa];Continue,{k,1,PrimePi[(Prime[n]-1)/2]}];Print[n," ",m];Continue,{n,1,80}]
%Y A236308 Cf. A000040, A000108, A234972, A235709, A235712, A236306, A236966.
%K A236308 nonn
%O A236308 1,9
%A A236308 _Zhi-Wei Sun_, Apr 21 2014