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%I #37 Aug 10 2017 14:27:01
%S 6,12,20,42,56,72,110,156,240,272,342,506,600,702,812,930,992,1332,
%T 1640,1806,2162,2352,2756,3422,3660,4032,4422,4970,5256,6162,6480,
%U 6806,7832,9312,10100,10506,11342,11772,12656,14520,15500,16002,16256,17030,18632
%N The order of the one-dimensional affine group in the finite fields F_q with q >= 3.
%C The affine group is the group of invertible affine transformations in F_q such as: x--> ax+b, a > 0.
%C These groups are Frobenius groups belonging to A178498
%C F_q is a field, so q = p^n, p is prime, with q >= 3 here.
%C The one-dimensional affine group in the finite fields F_q with q >= 3 is isomorphic to the semidirect product F_q x F_q^{*}, where F_q is endowed with the law +, and F_q^{*} is endowed with the law x. [_Bernard Schott_, Dec 22 2012]
%H Bernard Schott and Jean-Louis Tu, <a href="http://www.les-mathematiques.net/phorum/read.php?17,785127,785720#msg-785720">QDV8 & H62 : Hommage à Frobenius - Frobenius 8 - Exercice 8.2</a> (French mathematical forum les-mathematiques.net)
%H Jean-Pierre Serre, <a href="https://www.college-de-france.fr/media/jean-pierre-serre/UPL2937151343298039815_1___Groupes_finis.pdf">Groupes finis</a>, ENS - 1978/1979.
%F For the finite field F_q with q = p^n, the order of its affine group is q(q-1) = p^n(p^n-1), p prime, q >= 3.
%F a(n) = A000961(n+2)*((A000961(n+2)) - 1)). [_Bernard Schott_, Dec 22 2012]
%e a(1)=6 and this affine group of order 6 in the field F_3 is the dihedral group D_3 isomorphic to permutation group S_3.
%e a(2)=12 and this affine group of order 12 in the field F_4 is the semidirect product of Z(2) X Z(2) with Z(3).
%e a(6)=72 because for p=3, n=2 ==> q = p^n = 9 and 72 = q(q-1) = 9*8.
%p (p,n)-> p^n*(p^n-1)
%t mx = 20000; t = {}; p = 2; While[cnt = 0; n = 1; While[m = p^n (p^n - 1); m <= mx, AppendTo[t, m]; cnt++; n++]; cnt > 0, p = NextPrime[p]]; Union[Rest[t]] (* _T. D. Noe_, Dec 19 2012 *)
%Y Cf. A000961, A178498.
%K nonn
%O 1,1
%A _Bernard Schott_, Dec 07 2012
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