A206551 Moduli n for which the multiplicative group Modd n is cyclic.

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, 32, 33, 34, 35, 37, 38, 39, 41, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 67, 69, 71, 73, 74, 75, 77, 79, 81, 82, 83, 86, 87, 89, 93, 94, 95, 97, 98, 99

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.

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]

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

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.

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

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.

Examples

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

Crossrefs

Cf. A206550, A033948 (mod n case).

Formula

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