%I #22 Sep 13 2023 13:49:34

%S 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,19,21,22,23,25,26,27,29,31,

%T 32,33,34,35,37,38,39,41,43,45,46,47,49,50,51,53,54,55,57,58,59,61,62,

%U 64,67,69,71,73,74,75,77,79,81,82,83,86,87,89,93,94,95,97,98,99

%N Moduli n for which the multiplicative group Modd n is cyclic.

%C For Modd n (not to be confused with mod n) see a comment on A203571.

%C For n=1 one has the Modd 1 residue class [0], the integers. The group of order 1 is the cyclic group Z_1 with the unit element 0==1 (Modd 1). [Changed by _Wolfdieter Lang_, Apr 04 2012]

%C For the non-cyclic (acyclic) values see A206552.

%C For these numbers n, and only for these (only the n values < 100 are shown above), there exist primitive roots Modd n. See the nonzero values of A206550 for the smallest positive ones.

%C For n=1 the primitive root is 0 == 1 (Modd 1), see above.

%C For n>=1 the multiplicative group Modd n is the Galois group Gal(Q(rho(n))/Q), with the algebraic number rho(n) := 2*cos(Pi/n) with minimal polynomial C(n,x), whose coefficients are given in A187360.

%F A206550(a(n)) > 0, n>=1.

%e a(2) = 2 for the multiplicative group Modd 2, with representative [1], and there is a primitive root, namely 1, because 1^1 = 1 == 1 (Modd 1). The cycle structure is [[1]], the group is Z_1.

%e a(3) = 3 for the multiplicative group Modd 3 which coincides with the one for Modd 2.

%e a(4) = 4 for the multiplicative group Modd 4 with representatives [1,3]. The smallest positive primitive root is 3, because 3^2 == 1 (Modd 4). This group is cyclic, it is Z_2.

%Y Cf. A206550, A033948 (mod n case).

%K nonn

%O 1,2

%A _Wolfdieter Lang_, Mar 27 2012