cp's OEIS Frontend

A237112 Number of primes p < prime(n)/2 with p! a primitive root modulo prime(n).

%I A237112 #17 Aug 05 2019 02:48:56
%S A237112 0,0,1,0,2,2,1,1,1,2,2,2,2,2,4,4,6,4,2,3,3,4,7,9,5,7,5,8,4,7,6,7,8,7,
%T A237112 11,8,9,7,13,10,16,4,7,8,13,9,8,12,17,10,14,12,4,10,14,15,18,8,9,8,8,
%U A237112 18,6,8,7,16,9,11,21,15
%N A237112 Number of primes p < prime(n)/2 with p! a primitive root modulo prime(n).
%C A237112 Conjecture: a(n) > 0 for all n > 4. In other words, for any prime p > 7, there exists a prime q < p/2 such that q! is a primitive root modulo p.
%C A237112 See also A236306 for a similar conjecture.
%H A237112 Zhi-Wei Sun, <a href="/A237112/b237112.txt">Table of n, a(n) for n = 1..600</a>
%H A237112 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 A237112 a(7) = 1 since 3 is a prime smaller than prime(7)/2 = 17/2 and 3! = 6 is a primitive root modulo prime(7) = 17.
%e A237112 a(9) = 1 since  5 is a prime smaller than prime(9)/2 = 23/2 and 5! = 120 is a primitive root modulo prime(9) = 23.
%t A237112 f[k_]:=(Prime[k])!
%t A237112 dv[n_]:=Divisors[n]
%t A237112 Do[m=0;Do[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,1,70}]
%Y A237112 Cf. A000040, A000142, A234972, A235709, A235712, A236306, A236308, A236966, A237121.
%K A237112 nonn
%O A237112 1,5
%A A237112 _Zhi-Wei Sun_, Apr 22 2014