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

Author

Wolfdieter Lang, Aug 24 2012

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

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

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.