A207669 Numbers that match polynomials irreducible (mod 3), with coefficients in {0,1,2}.

3, 4, 5, 6, 7, 8, 10, 14, 17, 20, 22, 25, 34, 35, 38, 41, 43, 46, 49, 53, 58, 59, 65, 67, 71, 73, 77, 79, 86, 89, 92, 94, 97, 101, 110, 115, 118, 121, 125, 134, 137, 139, 145, 149, 151, 158, 166, 169, 172, 181, 185, 188, 190, 197, 205, 209, 212, 214, 217

Offset: 1

Author

Clark Kimberling, Feb 26 2012

Keywords

• nonn

Comments

For such polynomials irreducible over the field of rational numbers, see A207966, which also describes the enumeration of all the nonzero polynomials whose coefficients are all in {0,1,2}.

Examples

			Polynomials having coefficients in {0,1,2} are
enumerated by the positive integers as follows:
n ... p[n,x] .. irreducible (mod 3)
1 ... 1 ....... no
2 ... 2 ....... no
3 ... x ....... yes
4 ... x+1 ..... yes
5 ... x+2 ..... yes
6 ... 2x ...... yes
7 ... 2x+1 .... yes
8 ... 2x+2 .... yes
9 ... x^2 ..... no
10 .. x^2+1 ... yes
11 .. x^2+2 ... no
The least n for which p(n,x) is irreducible over the
rationals but not modulo 3 is 13; the factorization of
p(13,x) is (x+1)(x+2) (mod 3).
		

Crossrefs

Cf. A207670 (complement), A207671 (ternary).

Programs

Mathematica

t = Table[IntegerDigits[n, 3], {n, 1, 1000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
Table[p[n, x], {n, 1, 15}]
u = {}; Do[n++;
If[IrreduciblePolynomialQ[p[n, x], Modulus -> 3],
  AppendTo[u, n]], {n, 1, 400}]
u                           (* A207669 *)
Complement[Range[200], %]   (* A207670 *)
b[n_] := FromDigits[IntegerDigits[u, 3][[n]]]
Table[b[n], {n, 1, 50}]     (* A207671 *)