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

1,1

Comments

Church's table extends through degree 11.

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

Example

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.

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} ]

Prog

(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

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.

Sequence in context: A287626 A059458 A063697 * A222473 A361990 A335801

Adjacent sequences: A058940 A058941 A058942 * A058944 A058945 A058946

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Jan 13 2001

Status

approved