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A242242 Least positive primitive root g < prime(n) modulo prime(n) with 2^g - 1 also a primitive root modulo prime(n), or 0 if such a number g does not exist.

%I A242242 #9 Aug 05 2019 04:40:46
%S A242242 1,0,2,5,2,2,3,2,5,2,3,2,6,3,5,2,2,2,2,7,5,3,2,3,5,2,5,2,6,3,3,2,3,2,
%T A242242 2,6,5,2,5,2,2,2,19,5,2,3,2,3,2,6,3,7,7,6,3,5,2,6,5,3,3,2,5,17,10,2,3,
%U A242242 10,2,2
%N A242242 Least positive primitive root g < prime(n) modulo prime(n) with 2^g - 1 also a primitive root modulo prime(n), or 0 if such a number g does not exist.
%C A242242 Conjecture: a(n) > 0 except for n = 2. In other words, for any prime p > 3, there exists a primitive root 0 < g < p modulo p such that 2^g - 1 is also a primitive root modulo p.
%H A242242 Zhi-Wei Sun, <a href="/A242242/b242242.txt">Table of n, a(n) for n = 1..10000</a>
%H A242242 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1405.0290">Notes on primitive roots modulo primes</a>, arXiv:1405.0290 [math.NT], 2014.
%e A242242 a(4) = 5 since both 5 and 2^5 - 1 = 31 are primitive roots modulo prime(4) = 7, but none of 1, 2, 4 and 2^3 - 1 is a primitive root modulo prime(4) = 7.
%t A242242 f[n_]:=f[n]=2^n-1
%t A242242 dv[n_]:=Divisors[n]
%t A242242 Do[Do[If[Mod[f[k],Prime[n]]==0,Goto[aa]];Do[If[Mod[k^(Part[dv[Prime[n]-1],j])-1,Prime[n]]==0,Goto[aa]],{j,1,Length[dv[Prime[n]-1]]-1}];Do[If[rMod[f[k]^(Part[dv[Prime[n]-1],i])-1,Prime[n]]==0,Goto[aa]],{i,1,Length[dv[Prime[n]-1]]-1}];
%t A242242 Print[n," ",k];Goto[bb];Label[aa];Continue,{k,1,Prime[n]-1}];Label[cc];Print[n," ",0];Label[bb];Continue,{n,1,70}]
%Y A242242 Cf. A000040, A000225, A234972, A236966.
%K A242242 nonn
%O A242242 1,3
%A A242242 _Zhi-Wei Sun_, May 09 2014