A004124 - OEIS

%I M2383 #63 Oct 29 2023 10:51:00

%S 1,1,-3,-4,125,-3,-16807,256,-19683,125,-2357947691,144,1792160394037,

%T -16807,1265625,16777216,2862423051509815793,-19683,

%U -5480386857784802185939,4000000,205924456521,-2357947691,-39471584120695485887249589623,5308416

%N Discriminant of n-th cyclotomic polynomial.

%C 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.

%C 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

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

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

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

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

%H Gheorghe Coserea, <a href="/A004124/b004124.txt">Table of n, a(n) for n = 1..388</a> (first 100 terms from T. D. Noe)

%H Mohammad K. Azarian, <a href="http://ijpam.eu/contents/2007-36-2/9/9.pdf">On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials</a>, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537.  Zbl 1133.11012.

%H J. Shallit, <a href="/A004124/a004124.pdf">Letter to N. J. A. Sloane</a>, Mar 25 1980.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolynomialDiscriminant.html">Polynomial Discriminant</a>.

%F 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).

%F 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

%e a(100) = 2^40 * 5^70.

%e a(100) = ((-1)^(40*39/2))*(100^40)/(2^(40/1)*5^(40/4)) = +2^40*5^70. - _Wolfdieter Lang_, Aug 03 2011

%t 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}]

%t a[n_] := Discriminant[ Cyclotomic[n, x], x]; Table[a[n], {n, 1, 24}] (* _Jean-François Alcover_, Dec 06 2011 *)

%o (PARI) a(n) = poldisc(polcyclo(n));

%o (PARI)

%o a(n) = {

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

%o      g = prod(k=1, fsz, f[k,1]),

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

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

%o   return((1-2*((phi\2)%2)) * r);

%o };

%o vector(24, n, a(n))  \\ _Gheorghe Coserea_, Oct 31 2016

%K sign,easy,nice

%O 1,3

%A _N. J. A. Sloane_

%E Edited by _T. D. Noe_, Sep 30 2003