A058943 Coefficients of irreducible polynomials over GF(2) listed in lexicographic order.

10, 11, 111, 1011, 1101, 10011, 11001, 11111, 100101, 101001, 101111, 110111, 111011, 111101, 1000011, 1001001, 1010111, 1011011, 1100001, 1100111, 1101101, 1110011, 1110101, 10000011, 10001001, 10001111, 10010001

Offset: 1

Author

N. J. A. Sloane, Jan 13 2001

Keywords

• nonn
• easy
• nice

Comments

Church's table extends through degree 11.

Examples

			The first few are x, x+1; x^2+x+1; x^3+x+1, x^3+x^2+1; ... Note that x is irreducible but not primitive.
		

References

• R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, 1983, Table C, pp. 553-555.

Links

• T. D. Noe, Table of n, a(n) for n=1..1377 (through degree 13)
• R. Church, Tables of irreducible polynomials for the first four prime moduli, Annals Math., 36 (1935), 198-209.
• F. Ruskey, Irreducible and Primitive Polynomials over GF(2)
• Index entries for sequences containing GF(2)[X]-polynomials

Crossrefs

Cf. A000020, A001037, A011260, A058944-A058948.

Converted to decimal: A014580.

Irreducible over GF(2), GF(3), GF(4), GF(5), GF(7): this sequence, A058944, A058948, A058945, A058946. Primitive irreducible over GF(2), GF(3), GF(4), GF(5), GF(7): A058947, A058949, A058952, A058950, A058951.

Programs

Mathematica

Do[a = Reverse[ IntegerDigits[n, 2]]; b = {0}; l = Length[a]; k = 1; While[k < l + 1, b = Append[b, a[[k]]*x^(k - 1) ]; k++ ]; b = Apply[Plus, b]; c = Factor[b, Modulus -> 2]; If[b == c, Print[ FromDigits[ IntegerDigits[n, 2]]]], {n, 3, 250, 2} ]

PARI

seq(N, p=2, maxdeg=oo) = {
  my(a = List(), k=0, d=0);
  while (d++ <= maxdeg,
    for (n=p^d, 2*p^d-1, my(f=Mod(Pol(digits(n,p)),p));
      if(polisirreducible(f), listput(a, subst(lift(f),'x,10)); k++);
      if(k >= N, break(2))));
  Vec(a);
};
seq(27) \\ Gheorghe Coserea, May 28 2018