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A126270 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.

%I A126270 #12 Sep 07 2025 09:02:24
%S A126270 8,0,0,32,32,32,16,16,32,32,32,32,32,32,16,32,16,32,32,32,32,32,32,16,
%T A126270 32,32,32,32,32,32,16,32,32,32,0,32,32,32,32,32,32,32,32,32,32,32,32,
%U A126270 8,16,32,32,32,32,32,32,32,32,32,32,32,32,32,16,32,32,32,32,32,32,32,16
%N A126270 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.
%C A126270 P = 2*T_8(x/2), where T_8(x) is the degree 8 Chebyshev polynomial of the first kind.
%C A126270 For zeros in this sequence see A136362.
%H A126270 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html">Chebyshev Polynomial of the First Kind</a>
%e A126270 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.
%e A126270 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.
%o A126270 (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 */
%o A126270 (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 */
%Y A126270 Cf. A124827, A126271, A136362.
%K A126270 nonn,changed
%O A126270 0,1
%A A126270 _Artur Jasinski_, Dec 23 2006
%E A126270 Edited and extended by _Klaus Brockhaus_, Dec 27 2007