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Search: id:A004124
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| A004124 |
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Discriminant of n-th cyclotomic polynomial.
(Formerly M2383)
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+0
3
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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
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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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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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 91.
D. Marcus, Number Fields. Springer-Verlag, 1977, p. 27.
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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: (-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 pi^ei, a product of prime powers, a(n) = prod a(pi^ei)^phi(n/pi^ei).
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EXAMPLE
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a(100) = 2^40 5^70
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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}]
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PROGRAM
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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 this_sequence A077032 A041595 A041741
Adjacent sequences: A004121 A004122 A004123 this_sequence 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 (njas(AT)research.att.com).
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EXTENSIONS
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Edited by T. D. Noe (noe(AT)sspectra.com), Sep 30 2003
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