cp's OEIS Frontend

A237594 Number of primes p < prime(n)/2 such that the Bell number B(p) is a primitive root modulo prime(n).

%I A237594 #18 Aug 05 2019 03:04:05
%S A237594 0,0,1,1,2,1,2,3,1,1,2,4,2,4,6,5,7,3,4,3,4,2,12,7,3,5,4,9,5,6,4,5,12,
%T A237594 6,7,5,9,6,12,11,13,7,7,7,14,5,5,14,14,8,13,11,7,10,19,17,16,8,11,7,7,
%U A237594 23,11,12,10,22,14,8,22,11,20,22,13,13,15,24,27,14,18,18
%N A237594 Number of primes p < prime(n)/2 such that the Bell number B(p) is a primitive root modulo prime(n).
%C A237594 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 Bell number B(q) is a primitive root modulo p.
%H A237594 Zhi-Wei Sun, <a href="/A237594/b237594.txt">Table of n, a(n) for n = 1..500</a>
%H A237594 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 A237594 a(9) = 1 since 3 is a prime smaller than prime(9)/2 = 23/2 and B(3) = 5 is a primitive root modulo prime(9) = 23.
%t A237594 f[k_]:=BellB[Prime[k]]
%t A237594 dv[n_]:=Divisors[n]
%t A237594 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 A237594 Cf. A000040, A000110, A236308, A236966, A237112, A237121.
%K A237594 nonn
%O A237594 1,5
%A A237594 _Zhi-Wei Sun_, Apr 22 2014