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A193679 Sequence related to discriminant of cyclotomic polynomials A004124. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(p) = p for primes p.

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

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

EXAMPLE

n=6: a(6) = 2^(2/(2-1))*3^(2/(3-1)) = 12.

     Discriminant(Phi(6,x)) = -3 = - (6^phi(6))/a(6).

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-Fran├žois Alcover, Mar 21 2017 *)

PROG

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

CROSSREFS

Cf. A004124.

Sequence in context: A227987 A261863 A143482 * A066574 A240304 A325684

Adjacent sequences:  A193676 A193677 A193678 * A193680 A193681 A193682

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, Aug 20 2011

STATUS

approved

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Last modified January 25 23:04 EST 2020. Contains 331270 sequences. (Running on oeis4.)