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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.

%I A215041 #10 Dec 25 2023 18:06:42
%S A215041 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,
%T A215041 243,49,29,2025,31,2,2673,17,6125,729,37,19,9477,625,41,35721,43,121,
%U A215041 91125,23,47,6561,2401,3125,111537,169,53,19683,378125
%N 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.
%C A215041 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).
%C A215041 Compare this sequence with A193679, the anologon for the cyclotomic polynomials. See also the P. Ribenboim reference given in A004124.
%D A215041 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.
%F A215041 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).
%F A215041 a(1) = 1. Conjectures for a(n), n>=2: i) a(2^k) = 2, k>=1;
%F A215041   ii) a(p^k) = p^((p^(k-1)+1)/2), for odd prime p and k>=1;
%F A215041   iii) a(n) = product(p^(delta(n)/(p-1)), odd p|n) otherwise.
%e A215041 a(30) = 30^delta(30)/A193681(30) = 30^8/324000000 = 2025.
%e A215041 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;
%e A215041   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.
%Y A215041 Cf. A193681, A055034, A193679 (cyclotomic case).
%K A215041 nonn
%O A215041 1,2
%A A215041 _Wolfdieter Lang_, Aug 24 2012