A058947 Coefficients of primitive irreducible polynomials over GF(2) listed in lexicographic order.

11, 111, 1011, 1101, 10011, 11001, 100101, 101001, 101111, 110111, 111011, 111101, 1000011, 1011011, 1100001, 1100111, 1101101, 1110011, 10000011, 10001001, 10001111, 10010001, 10011101, 10100111, 10101011

Offset

1,1

Comments

Church's table extends through degree 11.

Links

T. D. Noe, Table of n, a(n) for n=1..1110 (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

Example

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

Mathematica

car = 2; maxDegree = 13;
okQ[{1, 1}] = True;
okQ[coefs_List] := Module[{P}, P = coefs.x^Range[Length[coefs]-1, 0, -1]; coefs[[1]] == 1 && IrreduciblePolynomialQ[P, Modulus -> car] && PrimitivePolynomialQ[P, car]];
FromDigits /@ Select[Table[IntegerDigits[k, car], {k, car+1, car^(maxDegree + 1)}], okQ] (* Jean-François Alcover, Sep 09 2019 *)

Crossrefs

Cf. A000020, A001037, A011260, A058943, A058944, A058945, A058946, A058948.

Irreducible over GF(2), GF(3), GF(4), GF(5), GF(7): A058943, A058944, A058948, A058945, A058946.

Primitive irreducible over GF(2), GF(3), GF(4), GF(5), GF(7): A058947, A058949, A058952, A058950, A058951.

a(n) = A007088(A091250(n)).

Sequence in context: A143573 A248039 A244204 * A282912 A284025 A283176

Adjacent sequences: A058944 A058945 A058946 * A058948 A058949 A058950

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Jan 13 2001

Status

approved