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A373946 Number of primitive polynomials of third degree over GF(m) with vanishing quadratic term with m = m(n) = A000961(n), for n >= 2.

%I A373946 #24 Jul 09 2024 01:55:11
%S A373946 1,1,0,4,3,18,8,16,18,48,48,27,80,48,108,108,72,300,144,224,180,308,
%T A373946 192,336,560,240,648,420,576,540,648,768,1080,1200,912,1360,1008,1352,
%U A373946 1188,1584,960,2340,1620,4410,2112,2432,1980,2952,1560,2592,2025,4592,2448,4872,4576
%N A373946 Number of primitive polynomials of third degree over GF(m) with vanishing quadratic term with m = m(n) = A000961(n), for n >= 2.
%C A373946 Apparently, a(n) = A373514(n) * A000010( 3 * A000961(n) - 3 ) * A025474(n) / 2, for n >= 2.
%H A373946 Martin Becker, <a href="/A373946/b373946.txt">Table of n, a(n) for n = 2..400</a>
%e A373946 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 A373946 (PARI)
%o A373946 is_max_o = (x1, x0, m, e)-> {
%o A373946   for(i = 1, #e, if(x1^e[i] == x0, return(0))); x1^m == x0;
%o A373946 }
%o A373946 count_them = (q)-> {
%o A373946   z = ffprimroot(ffgen(q, 'c));
%o A373946   m = q^3 - 1; f = factor(m); d = #f~;
%o A373946   e = vector(d, i, m/f[d + 1 - i, 1]);
%o A373946   co = vector(q - 1, i, z^(i - 1));
%o A373946   r = 0;
%o A373946   for(a = 1, q - 1,
%o A373946     for(b = 1, q - 1,
%o A373946       p = co[1]*x^3 + co[a]*x + co[b];
%o A373946       x1 = Mod(x, p); x0 = x1^0;
%o A373946       if(is_max_o(x1, x0, m, e) && polisirreducible(p), r += 1)
%o A373946     )
%o A373946   );
%o A373946   r;
%o A373946 }
%o A373946 print1(count_them(2));
%o A373946 for(q = 3, 64, if(isprimepower(q), print1(", ", count_them(q))))
%Y A373946 Cf. A000961, A319213, A373514.
%K A373946 nonn
%O A373946 2,4
%A A373946 _Martin Becker_, Jun 23 2024