A373946 - OEIS

%I #24 Jul 09 2024 01:55:11

%S 1,1,0,4,3,18,8,16,18,48,48,27,80,48,108,108,72,300,144,224,180,308,

%T 192,336,560,240,648,420,576,540,648,768,1080,1200,912,1360,1008,1352,

%U 1188,1584,960,2340,1620,4410,2112,2432,1980,2952,1560,2592,2025,4592,2448,4872,4576

%N Number of primitive polynomials of third degree over GF(m) with vanishing quadratic term with m = m(n) = A000961(n), for n >= 2.

%C Apparently, a(n) = A373514(n) * A000010( 3 * A000961(n) - 3 ) * A025474(n) / 2, for n >= 2.

%H Martin Becker, <a href="/A373946/b373946.txt">Table of n, a(n) for n = 2..400</a>

%e For n=5, m=5, there are 20 primitive polynomials over GF(5) of the form x^3+a*x^2+b*x+c. Among these, 4 polynomials have a=0: x^3+3*x+2, x^3+3*x+3, x^3+4*x+2, and x^3+4*x+3. Thus, a(5) = 4.

%o (PARI)

%o is_max_o = (x1, x0, m, e)-> {

%o   for(i = 1, #e, if(x1^e[i] == x0, return(0))); x1^m == x0;

%o }

%o count_them = (q)-> {

%o   z = ffprimroot(ffgen(q, 'c));

%o   m = q^3 - 1; f = factor(m); d = #f~;

%o   e = vector(d, i, m/f[d + 1 - i, 1]);

%o   co = vector(q - 1, i, z^(i - 1));

%o   r = 0;

%o   for(a = 1, q - 1,

%o     for(b = 1, q - 1,

%o       p = co[1]*x^3 + co[a]*x + co[b];

%o       x1 = Mod(x, p); x0 = x1^0;

%o       if(is_max_o(x1, x0, m, e) && polisirreducible(p), r += 1)

%o     )

%o   );

%o   r;

%o }

%o print1(count_them(2));

%o for(q = 3, 64, if(isprimepower(q), print1(", ", count_them(q))))

%Y Cf. A000961, A319213, A373514.

%K nonn

%O 2,4

%A _Martin Becker_, Jun 23 2024