A206552 Moduli n for which the multiplicative group Modd n is non-cyclic (acyclic).

12, 20, 24, 28, 30, 36, 40, 42, 44, 48, 52, 56, 60, 63, 65, 66, 68, 70, 72, 76, 78, 80, 84, 85, 88, 90, 91, 92, 96, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 126, 130, 132, 133, 136, 138, 140, 144, 145, 148, 150, 152, 154, 156, 160, 164, 165

Offset: 1

Author

Wolfdieter Lang, Mar 27 2012

Keywords

• nonn

Comments

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

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.

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.

Examples

			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).
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.
		

Crossrefs

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

Formula

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