cp's OEIS Frontend

A124827 Order of Galois groups of irreducible Chebyshev polynomials of order n.

%I A124827 #30 May 08 2025 08:54:46
%S A124827 1,2,6,8,20,12,42,16,54,40,110,48,156,84,120,64,272,108,342,160,252
%N A124827 Order of Galois groups of irreducible Chebyshev polynomials of order n.
%C A124827 All groups belonging to solvable Galois groups.
%C A124827 Very similar sequence is A002618 (disagreement occurred only for Chebyshev polynomials orders 8 and 16).
%C A124827 When the order of an irreducible Chebyshev polynomial is a prime number p, the Galois group is the Frobenius group of order p*(p-1) A036689.
%C A124827 In Magma classification the Galois groups are the following: T1_1, T2_1, T3_2, T4_3, T5_3, T6_3, T7_4, T8_8, T9_10, T11_4, T12_28, T13_6, T14_7, T15_11, T16_144, T17_5, T18_45, T19_6, T20_42, T21_15.
%C A124827 Is a(n) the order of Galois group of the polynomial x^n - 2? If so, then a(n) = n*phi(n) for n not divisible by 8, and n*phi(n)/2 otherwise (see the Math Overflow link below). Under this assumption, a(n) is multiplicative with a(p^e) = p^(2*e-1)*(p-1) for p being an odd prime; a(2) = 2, a(4) = 8, and a(2^e) = 2^(2*e-2) for e >= 3. - _Jianing Song_, Nov 22 2022
%H A124827 Math Overflow, <a href="https://mathoverflow.net/questions/143739/galois-group-of-xn-2">Galois Group of x^n - 2</a>
%e A124827 a(5)=20 because the order of the Galois group of polynomial 16x^5-20x^3+5x-c is 20 (where c is an integer chosen so that the polynomial is irreducible). This transitive group is the Frobenius group F5 of order 20 (also called the metacyclic group M_5) T5_3(20) in Magma classification.
%o A124827 (Magma) Zx<x>:=PolynomialRing(Integers()); f:=16*x^5-20*x^3+5*x-7; G:=GaloisGroup(f:Old); "Order of group",#G; // Juergen Klueners klueners(AT)math.uni-duesseldorf.de
%Y A124827 Cf. A001710, A000142, A036689, A002618, A036689.
%Y A124827 Cf. A127835.
%K A124827 nonn,uned,more
%O A124827 1,2
%A A124827 _Artur Jasinski_, Nov 09 2006