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

1,1

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}.

Links

Table of n, a(n) for n=1..59.

Example

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).

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 *)

Crossrefs

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

Sequence in context: A073632 A066378 A125684 * A001272 A273664 A364099

Adjacent sequences: A207666 A207667 A207668 * A207670 A207671 A207672

Keyword

nonn

Author

Clark Kimberling, Feb 26 2012

Status

approved