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A136415 Numbers n such that a type-3 Gaussian normal basis over GF(2^n) exists.

%I A136415 #19 Aug 30 2014 18:57:21
%S A136415 4,6,12,14,20,22,46,52,54,60,70,76,92,94,116,124,126,140,166,174,180,
%T A136415 182,204,206,214,220,230,236,244,252,262,276,284,286,292,294,302,332,
%U A136415 340,350,356,364,372,374,390,404,412,430,460,484,494,510,516,526,532
%N A136415 Numbers n such that a type-3 Gaussian normal basis over GF(2^n) exists.
%C A136415 A type-t Gaussian normal basis exists for GF(2^n) if p=n*t+1 is prime and gcd(n,(p-1)/ord(2 mod p))==1.
%C A136415 Type-1 basis correspond to sequence A071642, type-2 basis to A054639.
%H A136415 Joerg Arndt, Mar 31 2008, <a href="/A136415/b136415.txt">Table of n, a(n) for n = 1..201</a>
%H A136415 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 42.9  "Gaussian normal bases", pp.914-920
%e A136415 12 is in the list because 3*12+1=37 is prime and the index of 2 mod 37 (==36/ord(2 mod 37)==1, 2 is a generator mod 37) is coprime to 12.
%o A136415 (PARI)
%o A136415 gauss_test(n, t)=
%o A136415 { /* test whether a type-t Gaussian normal basis exists for GF(2^n) */
%o A136415   local( p, r2, g, d );
%o A136415   p = t*n + 1;
%o A136415   if ( !isprime(p), return( 0 ) );
%o A136415   if ( p<=2, return( 0 ) );
%o A136415   r2 = znorder( Mod(2, p) );
%o A136415   d = (p-1)/r2;
%o A136415   g = gcd(d, n);
%o A136415   return ( if ( 1==g, 1, 0) );
%o A136415 }
%o A136415 /* generate this sequence: */
%o A136415 t=3;ct=1;for(n=1,10^7,if(gauss_test(n,t), print1(n,", ");ct+=1;if(ct>200,break())))
%Y A136415 Cf. A071642, A054639.
%K A136415 nonn
%O A136415 1,1
%A A136415 _Joerg Arndt_, Mar 31 2008