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

Author

N. J. A. Sloane, Jan 13 2001

Keywords

• nonn
• easy
• nice

Examples

			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.
		

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.

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.

Programs

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 *)

Extensions

More terms from David Wasserman, May 23 2002