A207669 - OEIS

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

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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