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%I #6 Apr 25 2014 00:02:57
%S 1,2,2,3,2,2,3,2,5,2,3,2,7,3,5,2,2,2,2,7,5,3,2,3,5,2,5,2,11,3,3,2,3,2,
%T 2,7,5,2,5,2,2,2,22,5,2,3,2,3,2,7,3,7,7,11,3,5,2,15,5,3,3,2,5,22,15,2,
%U 3,15,2,2,3,7,11,2,2,5,2,5,3,22
%N Least positive primitive root g < prime(n) modulo prime(n) which is also a partition number given by A000041, or 0 if such a number g does not exist.
%C According to the conjecture in A241504, a(n) should be always positive.
%H Zhi-Wei Sun, <a href="/A241516/b241516.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.
%e a(4) = 3 since 3 = A000041(3) is a primitive root modulo prime(4) = 7, but neither 1 = A000041(1) nor 2 = A000041(2) is.
%t f[k_]:=PartitionsP[k]
%t dv[n_]:=Divisors[n]
%t Do[Do[If[f[k]>Prime[n]-1,Goto[cc]];Do[If[Mod[f[k]^(Part[dv[Prime[n]-1],i]),Prime[n]]==1,Goto[aa]],{i,1,Length[dv[Prime[n]-1]]-1}];Print[n," ",PartitionsP[k]];Goto[bb];Label[aa];Continue,{k,1,Prime[n]-1}];Label[cc];Print[Prime[n]," ",0];Label[bb];Continue,{n,1,80}]
%Y Cf. A000040, A000041, A237121, A239957, A239963, A241476, A241492, A241504.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Apr 24 2014
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