A215041 a(n) = n^degree(C(n,x))/discriminant(C(n,x)) for the minimal polynomials C(n,x) of 2*cos(Pi/n), given in A187360.

1, 2, 3, 2, 5, 3, 7, 2, 9, 5, 11, 9, 13, 7, 45, 2, 17, 27, 19, 25, 189, 11, 23, 81, 125, 13, 243, 49, 29, 2025, 31, 2, 2673, 17, 6125, 729, 37, 19, 9477, 625, 41, 35721, 43, 121, 91125, 23, 47, 6561, 2401, 3125, 111537, 169, 53, 19683, 378125

Offset

1,2

Comments

The discriminants for C(n,x), the minimal polynomial of 2*cos(Pi/n) are found under A193681. The degree of C(n,x), called delta(n), is given as A055034(n).

Compare this sequence with A193679, the anologon for the cyclotomic polynomials. See also the P. Ribenboim reference given in A004124.

References

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.

Links

Table of n, a(n) for n=1..55.

Formula

a(n) = (n^delta(n))/Discriminant(C(n,x)), n>=1, with the minimal polynomials C(n,x) of 2*cos(Pi/n), with coefficient triangle given in A187360, and their degree delta(n) given in A055034(n).

a(1) = 1. Conjectures for a(n), n>=2: i) a(2^k) = 2, k>=1;

ii) a(p^k) = p^((p^(k-1)+1)/2), for odd prime p and k>=1;

iii) a(n) = product(p^(delta(n)/(p-1)), odd p|n) otherwise.

Example

a(30) = 30^delta(30)/A193681(30) = 30^8/324000000 = 2025.
For the conjectures: i) a(4) = 2; ii) a^(3^2) = a(9) = 3^((3+1)/2) = 9; iii) a(30) = a(2*3*5) = 3^(delta(30)/2)*5^(delta(30)/4) = 3^4*5^2 = 2025;
  a(40) = a(2^3*5) = 5^(delta(40)/4) = 5^4 = 625; a(45) = a(3^2*5) = 3^(delta(45)/2)* 5^(delta(45)/4) = 91125.

Crossrefs

Cf. A193681, A055034, A193679 (cyclotomic case).

Sequence in context: A192330 A320028 A327076 * A347241 A338668 A284695

Adjacent sequences: A215038 A215039 A215040 * A215042 A215043 A215044

Keyword

nonn

Author

Wolfdieter Lang, Aug 24 2012

Status

approved