A193679 Sequence related to discriminant of cyclotomic polynomials A004124.

1, 2, 3, 4, 5, 12, 7, 16, 27, 80, 11, 144, 13, 448, 2025, 256, 17, 1728, 19, 6400, 35721, 11264, 23, 20736, 3125, 53248, 19683, 200704, 29, 518400, 31, 65536, 7144929, 1114112, 37515625, 2985984, 37, 4980736, 89813529, 40960000, 41, 146313216, 43, 126877696

Offset: 1

Author

Wolfdieter Lang, Aug 20 2011

Keywords

• nonn

Comments

a(p) = p for primes p.

Examples

			n=6: a(6) = 2^(2/(2-1))*3^(2/(3-1)) = 12.
     Discriminant(Phi(6,x)) = -3 = - (6^phi(6))/a(6).
		

References

• P. Ribenboim, Classical Theory of Algebraic Numbers, Springer, 2001, p. 297, eq.(1).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• 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.
• Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Crossrefs

Cf. A004124.

Programs

Maple

with(numtheory): A193679 := n -> n^phi(n)/abs(discrim(cyclotomic (n,x),x)); seq(A193679(i),i=1..49); # Peter Luschny, Aug 20 2011

Mathematica

a[n_] := n^EulerPhi[n]/Abs[Discriminant[Cyclotomic[n, x], x]]; Array[a, 44]
(* Jean-François Alcover, Mar 21 2017 *)
Table[Product[d^(-n*MoebiusMu[d]/d), {d, Divisors[n]}], {n, 1, 50}] (* Vaclav Kotesovec, May 12 2024 *)
Table[Product[p^(EulerPhi[n]/(p-1)), {p, Select[Divisors[n], PrimeQ[#]&]}], {n, 1, 50}] (* Vaclav Kotesovec, May 13 2024 *)

PARI

a(n) = n^eulerphi(n)/abs(poldisc(polcyclo(n))); \\ Michel Marcus, Jul 14 2018

Formula

a(n) = n^phi(n)/abs(discriminant(Phi(n,x))), n>=1, with the cyclotomic polynomials Phi(n,x) and the Euler totient function phi(n)=A000010(n).

a(n) = product(p^(phi(n)/(p-1)),p prime dividing n), n>=2, a(1)=1.

Conjecture: Dirichlet g.f. of log(a(n)): -zeta(s-1)*zeta'(s)/zeta(s)^2, where zeta'(s) is the derivative of zeta(s). This would give a(n) = exp(Sum_{d|n} Lambda(d)*phi(n/d)), with Lambda(n)=log(A014963) and phi(n)=A000010. - Benedict W. J. Irwin, Jul 14 2018