A206552 - OEIS

%I #14 Dec 27 2012 11:41:41

%S 12,20,24,28,30,36,40,42,44,48,52,56,60,63,65,66,68,70,72,76,78,80,84,

%T 85,88,90,91,92,96,100,102,104,105,108,110,112,114,116,117,120,124,

%U 126,130,132,133,136,138,140,144,145,148,150,152,154,156,160,164,165

%N Moduli n for which the multiplicative group Modd n is non-cyclic (acyclic).

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

%C Precisely these numbers n (only the ones <=165 are shown above) have no primitive root Modd n. See the zero entries of A206550, except A206550(1) = 0 which stands for a primitive root 0.

%C The multiplicative Modd n group 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(1) = 12 because A206550(12) = 0 for the first time, not counting A206550(1) = 0. The cycle structure of the multiplicative Modd 12 group is [[5,1],[7,1],[11,1]]. This is the (abelian, non-cyclic) group Z_2 x Z_2 (isomorphic to the Klein group V_4 or Dih_2).

%e a(2) = 20 because A206550(20) = 0 for the second time, not counting A206550(1) = 0. The cycle structure of the multiplicative Modd 20 group is [[3,9,13,1],[7,9,17,1],[11,1],[19,1]]. This is the (abelian, non-cyclic) group Z_4 x Z_2.

%Y Cf. A206550, A206551, A033949 (mod n case).

%K nonn

%O 1,1

%A _Wolfdieter Lang_, Mar 27 2012