A058944 Coefficients of monic irreducible polynomials over GF(3) listed in lexicographic order.

10, 11, 12, 101, 112, 122, 1021, 1022, 1102, 1112, 1121, 1201, 1211, 1222, 10012, 10022, 10102, 10111, 10121, 10202, 11002, 11021, 11101, 11111, 11122, 11222, 12002, 12011, 12101, 12112, 12121, 12212, 100021, 100022, 100112, 100211

Offset

1,1

References

R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, 1983, Table C, pp. 555-557.

Links

Robert Israel and T. D. Noe, Table of n, a(n) for n = 1..10000 (n = 1..1318 from T. D. Noe)

R. Church, Tables of irreducible polynomials for the first four prime moduli, Annals Math., 36 (1935), 198-209. Church's table extends through degree 7.

Example

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

Maple

N:= 100: # to get the first N terms
count:= 0:
for d from 1 while count < N do
for nn from 0 to 3^d-1 while count < N do
  L:= convert(nn, base, 3);
  P:= add(L[i]*x^(i-1), i=1..nops(L))+x^d;
  if Irreduc(P) mod 3 then
     count:= count+1;
     A[count]:= add(L[i]*10^(i-1), i=1..nops(L))+10^d;
  fi
  od
od:
seq(A[i], i=1..N); # Robert Israel, Jul 06 2016

Mathematica

A058944 = Union[ Reap[ Do[ a = Reverse[ IntegerDigits[n, 3]]; b = {0}; la = Length[a]; k = 1; While[k < la+1, b = Append[ b, a[[k]]*x^(k-1)]; k++]; b = Plus @@ b; c = Factor[ b, Modulus -> 3]; id = IntegerDigits[n, 3]; If[b == c && (id == {1, 0} || id[[-1]] != 0), Sow[ FromDigits[id] ] ], {n, 3, 300}]][[2, 1]]](* Jean-François Alcover, Feb 13 2012, after A058943 *)

Crossrefs

Cf. A058943, A058945-A058948.

Irreducible over GF(2), GF(3), GF(4), GF(5), GF(7): A058943, this sequence, A058948, A058945, A058946. Primitive irreducible over GF(2), GF(3), GF(4), GF(5), GF(7): A058947, A058949, A058952, A058950, A058951.

Sequence in context: A239903 A244159 A171752 * A167178 A077677 A216502

Adjacent sequences: A058941 A058942 A058943 * A058945 A058946 A058947

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Jan 13 2001

Extensions

More terms from David Wasserman, May 23 2002

Status

approved