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A004124
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Discriminant of n-th cyclotomic polynomial.
(Formerly M2383)
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7
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1, 1, -3, -4, 125, -3, -16807, 256, -19683, 125, -2357947691, 144, 1792160394037, -16807, 1265625, 16777216, 2862423051509815793, -19683, -5480386857784802185939, 4000000, 205924456521, -2357947691, -39471584120695485887249589623, 5308416
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OFFSET
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1,3
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COMMENTS
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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.
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REFERENCES
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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).
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..100
Eric Weisstein's World of Mathematics, Polynomial Discriminant
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FORMULA
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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. [From Wolfdieter Lang, Aug 03 2011]
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EXAMPLE
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a(100) = 2^40 5^70.
a(100) = ((-1)^(40*39/2))*(100^40)/(2^(40/1)*5^(40/4)) = +2^40*5^70. [From Wolfdieter Lang, Aug 03 2011]
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MATHEMATICA
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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}] (* From Jean-François Alcover, Dec 06 2011 *)
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PROG
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(PARI) for(n=1, 30, print(poldisc(polcyclo(n))))
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CROSSREFS
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Sequence in context: A041351 A066496 A041465 * A175504 A077032 A041595
Adjacent sequences: A004121 A004122 A004123 * A004125 A004126 A004127
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KEYWORD
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sign,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Edited by T. D. Noe, Sep 30 2003
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STATUS
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approved
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