A126270 -id:A126270 - cp's OEIS frontend

A126271 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) = 128x^8 - 256x^6 + 160x^4 - 32x^2 + 1.

32, 32, 16, 32, 32, 32, 32, 32, 32, 16, 32, 32, 32, 16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 16, 16, 32, 32, 32

Offset: 0

Artur Jasinski, Dec 23 2006

nonn

P(x) = T_8(x) is the degree 8 Chebyshev polynomial of the first kind.

Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the First Kind

Cf. A124827, A126270.

Q:=RationalField(); R:=PolynomialRing(Q); f:=128*x^8 - 256*x^6 + 160*x^4 - 32*x^2 + 0; for n in {0 .. 30} do f:=f+1; Order(GaloisGroup(f)); end for; /* N. J. A. Sloane */

Edited by N. J. A. Sloane, Dec 28 2007

A136362 Numbers n such that P+n is not irreducible, where P = x^8 - 8x^6 + 20x^4 - 16*x^2 + 2.

Original entry on oeis.org

1, 2, 34, 254, 898, 2302, 4898, 9214, 15874, 25598, 39202, 57598, 81794, 112894, 152098, 200702, 260098, 331774, 417314, 518398, 636802, 774398, 933154, 1115134, 1322498

Offset: 1

Klaus Brockhaus, Dec 27 2007

nonn

P = 2*(substitution of x by x/2 in T_8(x)), where T_8(x) is degree 8 Chebyshev polynomial of the first kind.

			P+254 = x^8 - 8*x^6 + 20*x^4 - 16*x^2 + 256 = (x^4 - 10*x^2 + 32)*(x^4 + 2*x^2 + 8).

Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the First Kind

Cf. A126270.

Zx:= PolynomialRing(Integers()); T:=Coefficients(ChebyshevT(8)); P:=Zx ! [ 2^(2-i)*T[i]: i in [1..#T] ]; [ n: n in [0..1340000] | not IsIrreducible(P+n) ];

a(1) = 1; a(2) = 2; for n > 2, a(n) = 4*n^2*(n-2)^2-2.

G.f.: x*(4*x^6 - 21*x^5 + 47*x^4 - 94*x^3 - 34*x^2 + 3*x - 1)/(x - 1)^5.

cp's OEIS Frontend

A126271 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) = 128x^8 - 256x^6 + 160x^4 - 32x^2 + 1.

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A136362 Numbers n such that P+n is not irreducible, where P = x^8 - 8x^6 + 20x^4 - 16*x^2 + 2.

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cp's OEIS Frontend

A126271 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) = 128*x^8 - 256*x^6 + 160*x^4 - 32*x^2 + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Extensions

A136362 Numbers n such that P+n is not irreducible, where P = x^8 - 8*x^6 + 20*x^4 - 16*x^2 + 2.

Views

Author

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Comments

Examples

Links

Crossrefs

Programs

Magma

Formula

A126271 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) = 128x^8 - 256x^6 + 160x^4 - 32x^2 + 1.

A136362 Numbers n such that P+n is not irreducible, where P = x^8 - 8x^6 + 20x^4 - 16*x^2 + 2.