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A004124 Discriminant of n-th cyclotomic polynomial.
(Formerly M2383)
8
1, 1, -3, -4, 125, -3, -16807, 256, -19683, 125, -2357947691, 144, 1792160394037, -16807, 1265625, 16777216, 2862423051509815793, -19683, -5480386857784802185939, 4000000, 205924456521, -2357947691, -39471584120695485887249589623, 5308416 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

n and a(n) have the same prime factors, except when 2 divides n but 4 does not divide n, then n/2 and a(n) have the same prime factors.

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.

E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 91.

D. Marcus, Number Fields. Springer-Verlag, 1977, p. 27.

P. Ribenboim, Classical Theory of Algebraic Numbers, Springer, 2001, pp. 118-9 and p. 297.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe and Gheorghe Coserea, Table of n, a(n) for n = 1..388 (first 100 terms from T. D. Noe)

Eric Weisstein's World of Mathematics, Polynomial Discriminant

J. Shallit, Letter to N. J. A. Sloane, Mar 25 1980

FORMULA

Sign(a(n)) = (-1)^(phi(n)*(phi(n)-1)/2). Magnitude: For prime p, a(p) = p^(p-2). For n = p^e, a prime power, a(n) = p^(((p-1)e-1) p^(e-1)). For n = prod p_i^e_i, a product of prime powers, a(n) = prod a(p_i^e_i)^phi(n/p_i^e_i).

a(n) = Sign(a(n))*(n^phi(n))/product(p^(phi(n)/(p-1)),p|n). See the Ribenboim reference, p. 297, eq.(1), with the sign taken from the previous formula and n=2 included. [Wolfdieter Lang, Aug 03 2011]

EXAMPLE

a(100) = 2^40 5^70.

a(100) = ((-1)^(40*39/2))*(100^40)/(2^(40/1)*5^(40/4))  = +2^40*5^70. [Wolfdieter Lang, Aug 03 2011]

MATHEMATICA

PrimePowers[n_] := Module[{f, t}, f=FactorInteger[n]; t=Transpose[f]; First[t]^Last[t]]; app[pp_] := Module[{f, p, e}, f=FactorInteger[pp]; p=f[[1, 1]]; e=f[[1, 2]]; p^(((p-1)e-1) p^(e-1))]; SetAttributes[app, Listable]; a[n_] := Module[{pp, phi=EulerPhi[n]}, If[n==1, 1, pp=PrimePowers[n]; (-1)^(phi*(phi-1)/2) Times@@(app[pp]^EulerPhi[n/pp])]]; Table[a[n], {n, 24}]

a[n_] := Discriminant[ Cyclotomic[n, x], x]; Table[a[n], {n, 1, 24}] (* Jean-Fran├žois Alcover, Dec 06 2011 *)

PROG

(PARI) for(n=1, 30, print(poldisc(polcyclo(n))))

(PARI)

a(n) = {

  my(f = factor(n), fsz = matsize(f)[1],

     g = prod(k=1, fsz, f[k, 1]),

     h = prod(k=1, fsz, f[k, 1]-1), phi = (n\g)*h,

     r = prod(k=1, fsz, f[k, 1] ^ ((phi\(f[k, 1]-1)) * (f[k, 2]*(f[k, 1]-1)-1))));

  return((1-2*((phi\2)%2)) * r);

};

vector(24, n, a(n))  \\ Gheorghe Coserea, Oct 31 2016

CROSSREFS

Sequence in context: A266516 A066496 A041465 * A175504 A280735 A290282

Adjacent sequences:  A004121 A004122 A004123 * A004125 A004126 A004127

KEYWORD

sign,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Edited by T. D. Noe, Sep 30 2003

STATUS

approved

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Last modified April 19 06:30 EDT 2019. Contains 322237 sequences. (Running on oeis4.)