Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #10 Dec 25 2023 18:06:42
%S 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 243,49,29,2025,31,2,2673,17,6125,729,37,19,9477,625,41,35721,43,121,
%U 91125,23,47,6561,2401,3125,111537,169,53,19683,378125
%N 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 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 Compare this sequence with A193679, the anologon for the cyclotomic polynomials. See also the P. Ribenboim reference given in A004124.
%D 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 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 a(1) = 1. Conjectures for a(n), n>=2: i) a(2^k) = 2, k>=1;
%F ii) a(p^k) = p^((p^(k-1)+1)/2), for odd prime p and k>=1;
%F iii) a(n) = product(p^(delta(n)/(p-1)), odd p|n) otherwise.
%e a(30) = 30^delta(30)/A193681(30) = 30^8/324000000 = 2025.
%e 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 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 Cf. A193681, A055034, A193679 (cyclotomic case).
%K nonn
%O 1,2
%A _Wolfdieter Lang_, Aug 24 2012