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

1,1

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).

Links

Table of n, a(n) for n=1..47.

Formula

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

Example

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.

Sequence in context: A010381 A095033 A010399 * A008523 A047568 A139453

Adjacent sequences: A215043 A215044 A215045 * A215047 A215048 A215049

Keyword

nonn

Author

Wolfdieter Lang, Sep 03 2012

Status

approved