%I #19 Aug 30 2014 18:57:21

%S 4,6,12,14,20,22,46,52,54,60,70,76,92,94,116,124,126,140,166,174,180,

%T 182,204,206,214,220,230,236,244,252,262,276,284,286,292,294,302,332,

%U 340,350,356,364,372,374,390,404,412,430,460,484,494,510,516,526,532

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

%C 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 Type-1 basis correspond to sequence A071642, type-2 basis to A054639.

%H Joerg Arndt, Mar 31 2008, <a href="/A136415/b136415.txt">Table of n, a(n) for n = 1..201</a>

%H 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 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 (PARI)

%o gauss_test(n, t)=

%o { /* test whether a type-t Gaussian normal basis exists for GF(2^n) */

%o local( p, r2, g, d );

%o p = t*n + 1;

%o if ( !isprime(p), return( 0 ) );

%o if ( p<=2, return( 0 ) );

%o r2 = znorder( Mod(2, p) );

%o d = (p-1)/r2;

%o g = gcd(d, n);

%o return ( if ( 1==g, 1, 0) );

%o }

%o /* generate this sequence: */

%o t=3;ct=1;for(n=1,10^7,if(gauss_test(n,t), print1(n,", ");ct+=1;if(ct>200,break())))

%Y Cf. A071642, A054639.

%K nonn

%O 1,1

%A _Joerg Arndt_, Mar 31 2008