A206550 Smallest positive primitive roots Modd n.

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, 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, 7, 0, 3, 0, 5, 5, 13, 0, 3, 0, 3, 0, 5, 7, 3, 0, 0, 5, 11

Offset: 1

Author

Wolfdieter Lang, Mar 27 2012

Keywords

• nonn

Comments

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

The 0 for n=1 is a primitive root Modd 1, the other zeros indicate that there is no primitive root for this n.

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]

Examples

			n=1: delta(1) = 1, a(1) = 1 == 0 (Modd 1): 0^1 = 0 == 1 (Modd 1).
n=2: delta(2) = 1, a(2) = 1 == 1 (Modd 2): 1^1 = 1 == 1 (Modd 2).
n=4: delta(4) = 2, a(2) = 3 == 3 (Modd 4): 3^2 = 9 == 1 (Modd 4).
n=6: delta(4) = 2, a(6) = 5 == 5 (Modd 6): 5^2 = 25. 25 (Modd 6) = 25 (mod 6) =1.
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.
		

Crossrefs

Cf. A046145 (mod n case).

Formula

a(1) = 0 == 1 (Modd 1).

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.