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%I #23 Oct 11 2021 18:29:50
%S 3,5,5,7,3,5,5,19,3,7,5,6,17,7,6,5,3,13,3,5,7,3,5,11,5,3,3,11,5,5,5,5,
%T 6,14,3,3,3,17,5,3,3,6,13,5,7,3,5,11,5,19,3,5,5,3,6,10,5,5,14,6,3,7,5,
%U 5,7,5,3,3,11,5,5,3,5,6,7,3,5,7,3,7,5,5,5,17
%N Least positive primitive root of A139035(n).
%C If p is a prime, then a is called a semiprimitive root if it has order (p-1)/2 and there is no x for which a^x is congruent to -1 (mod p). So +- a^k, 0 <= k <= (p-3)/2 is a complete set of nonzero residues (mod p). A primitive root has order p-1, so a number cannot be both a primitive root and a semiprimitive root.
%C A139035 are the primes for which 2 is a semiprimitive root. This sequence gives the smallest positive primitive root corresponding to each term of A139035, so each term is greater than or equal to 3.
%H Amiram Eldar, <a href="/A179858/b179858.txt">Table of n, a(n) for n = 1..10000</a>
%e Since A139035(13)=311, 2 is a semiprimitive root of 311 so j=0,...,154, {+-2^j} is a complete set of residues (congruent to {1,...,310}). The corresponding member of this sequence is a(13)=17 because 17 is the smallest positive integer a for which {a^k}, k=0,...,309 is a complete set of residues.
%t PrimitiveRoot /@ Reap[For[p = 3, p < 3000, p = NextPrime[p], rp = MultiplicativeOrder[2, p]; rm = MultiplicativeOrder[-2, p]; If[rp != p-1 && rm == p-1, Sow[p]]]][[2, 1]] (* _Jean-François Alcover_, Sep 03 2016, after _Joerg Arndt_'s code for A139035 *)
%Y Cf. A001918, A139035.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Jan 11 2011
%E More terms from _Jean-François Alcover_, Sep 03 2016
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