A004124 Discriminant of n-th cyclotomic polynomial. (Formerly M2383)

1, 1, -3, -4, 125, -3, -16807, 256, -19683, 125, -2357947691, 144, 1792160394037, -16807, 1265625, 16777216, 2862423051509815793, -19683, -5480386857784802185939, 4000000, 205924456521, -2357947691, -39471584120695485887249589623, 5308416

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.

a(n) is negative <=> phi(n) == 2 (mod 4) <=> n = 4 or n is of the form p^e or 2*p^e, where p is a prime congruent to 3 modulo 4. - Jianing Song, May 17 2021

References

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

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

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537. Zbl 1133.11012.

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

Eric Weisstein's World of Mathematics, Polynomial Discriminant.

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 = Product_{i=1..k} p_i^e_i, a product of prime powers, a(n) = Product_{i=1..k} a(p_i^e_i)^phi(n/p_i^e_i).

a(n) = Sign(a(n))*(n^phi(n))/(Product_{p|n, p prime} p^(phi(n)/(p-1))). 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-François Alcover, Dec 06 2011 *)

Prog

(PARI) a(n) = 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 A308944 A280735

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