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A126270
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a(n) = order of Galois group of the polynomial P(x) + n if P(x) + n (after dividing by the gcd of its coefficients) is irreducible, otherwise a(n) = 0, where P(x) = x^8 - 8*x^6 + 20*x^4 - 16*x^2 + 2.
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2
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8, 0, 0, 32, 32, 32, 16, 16, 32, 32, 32, 32, 32, 32, 16, 32, 16, 32, 32, 32, 32, 32, 32, 16, 32, 32, 32, 32, 32, 32, 16, 32, 32, 32, 0, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 8, 16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 16, 32, 32, 32, 32, 32, 32, 32, 16
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OFFSET
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0,1
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COMMENTS
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P = 2*T_8(x/2), where T_8(x) is the degree 8 Chebyshev polynomial of the first kind.
For zeros in this sequence see A136362.
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LINKS
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EXAMPLE
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Galois group of P+6 = x^8 - 8*x^6 + 20*x^4 - 16*x^2 + 8 is group 6 of degree 8 in MAGMA Transitive Group Identification, permutation group acting on a set of cardinality 8 with two generators, isomorphic to dihedral group D(8); it has index 2520 in symmetric group Sym(8) and order 16. Hence a(6) = 16.
a(34) = 0 since P+34 = x^8 - 8*x^6 + 20*x^4 - 16*x^2 + 36 = (x^4 - 8*x^2 + 18)*(x^4 + 2) is not irreducible.
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PROG
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(Magma) Zx<x>:=PolynomialRing(Integers()); T:=Coefficients(ChebyshevT(8)); P:=Zx ! [ 2^(2-i)*T[i]: i in [1..#T] ]; [ IsIrreducible(f) select Order(GaloisGroup(f)) else 0 where f is P+n: n in [0..70] ]; /* Klaus Brockhaus, Dec 27 2007 */
(Magma) Q:=RationalField(); R<x>:=PolynomialRing(Q); f:=x^8 - 8*x^6 + 20*x^4 - 16*x^2 + 1; for n in {0 .. 30} do f:=f+1; if IsIrreducible(f) then Order(GaloisGroup(f)); else 0; end if; end for; /* N. J. A. Sloane, Dec 28 2007 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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