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

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

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

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)).

Programs

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 *)