A004124 Discriminant of n-th cyclotomic polynomial.

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

Author

N. J. A. Sloane

Keywords

• sign
• easy
• nice

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

Examples

			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
		

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.

Programs

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 *)

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

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

Extensions

Edited by T. D. Noe, Sep 30 2003