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A206550 Smallest positive primitive roots Modd n.

%I A206550 #12 Aug 09 2022 14:17:11
%S A206550 0,1,1,3,3,5,3,3,5,3,3,0,7,5,7,3,3,5,3,0,11,3,3,0,3,7,5,0,3,0,3,3,5,3,
%T A206550 3,0,5,13,7,0,7,0,3,0,7,3,3,0,3,3,5,0,3,5,3,0,5,3,3,0,7,7,0,3,0,0,7,0,
%U A206550 7,0,3,0,5,5,13,0,3,0,3,0,5,7,3,0,0,5,11
%N A206550 Smallest positive primitive roots Modd n.
%C A206550 For multiplication Modd n (not to be confused with mod n) see a comment on A203571.
%C A206550 The 0 for n=1 is a primitive root Modd 1, the other zeros indicate that there is no primitive root for this n.
%C A206550 Iff a(n)>0, for n>=2, then the Galois group Gal(Q(2*cos(Pi/n))/Q), which is the multiplicative group of odd reduced residue classes Modd n (hence the notation Modd) is cyclic. For n=1 this group is also cyclic. See A206551 (cyclic moduli n) and A206552 (acyclic, i.e. non-cyclic, moduli n). [Changed by _Wolfdieter Lang_, Apr 04 2012]
%F A206550 a(1) = 0 == 1 (Modd 1).
%F A206550 If no primitive root exists for n>=2 then a(n):=0. If a primitive root exists for n>=2 then a(n) is  the smallest positive integer whose order Modd n is delta(n), with delta(n) = A055034(n). That is, with gcd(a(n),2*n) = 1, n>=2, the least positive exponent k such that a(n)^k == 1 (Modd n) is delta(n), and a(n) is the smallest  positive representative Modd n with this property.
%e A206550 n=1: delta(1) = 1, a(1) = 1 == 0 (Modd 1): 0^1 = 0 == 1 (Modd 1).
%e A206550 n=2: delta(2) = 1, a(2) = 1 == 1 (Modd 2): 1^1 = 1 == 1 (Modd 2).
%e A206550 n=4: delta(4) = 2, a(2) = 3 == 3 (Modd 4): 3^2 = 9 == 1 (Modd 4).
%e A206550 n=6: delta(4) = 2, a(6) = 5 == 5 (Modd 6): 5^2 = 25. 25 (Modd 6) = 25 (mod 6) =1.
%e A206550 n=12: delta(12) = 4, a(12) = 0, because no primitive root exists: 5^2 == 1 (Modd 12), 7^2 == 1 (Modd 12) and 11^2 == 1 (Modd 12). The cycle structure of this acyclic group is [[5,1],[7,1],[11,1]]. It is the (abelian) group Z_2 x Z_2.
%Y A206550 Cf. A046145 (mod n case).
%K A206550 nonn
%O A206550 1,4
%A A206550 _Wolfdieter Lang_, Mar 27 2012