A215046 Increasingly ordered list of those values m for which the degree of the minimal polynomial of 2*cos(Pi/m) (see A187360) is prime.

4, 5, 6, 7, 9, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459

Offset: 1

Author

Wolfdieter Lang, Sep 03 2012

Keywords

• nonn

Comments

The degree delta(m) of the minimal polynomial of rho(m) := 2*cos(Pi/m), called C(m,x) with coefficient array A187360, is given by A055034(m).

If delta(m) = phi(2*m)/2, m>=2, delta(1) = 1, with phi = A000010, is prime then the (Abelian) Galois group G(Q(rho(m))/Q) is cyclic. Because this Galois group of C(m,x) has order delta(m) this follows from a corollary to Lagrange's theorem, or also from Cauchy's theorem on groups.

Because the mentioned Galois group is isomorphic to the multiplicative group Modd m of order delta(m) (see a comment on A203571) all m = a(n) values appear in A206551.

This sequence is also a subsequence of A210845 because p is squarefree (see A005117).

Examples

			a(4) = 7, because 7 satisfies phi(14)/2 = phi(2*7)/2 = 1*6/2 = 3, which is prime; and 7 is the fourth smallest number m satisfying: phi(2*m)/2 is prime.
		

Crossrefs

Cf. A055034.

Formula

phi(2*m)/2 is prime iff m=a(n), n>=1, with phi = A000010 (Euler's totient).