A207671 -id:A207671 - cp's OEIS frontend

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

Clark Kimberling, Feb 26 2012

nonn

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}.

			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).

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

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 *)

A207670 Numbers that match polynomials with coefficients in {0,1,2} that are not irreducible modulo 3; complement of A207670.

Original entry on oeis.org

1, 2, 9, 11, 12, 13, 15, 16, 18, 19, 21, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 36, 37, 39, 40, 42, 44, 45, 47, 48, 50, 51, 52, 54, 55, 56, 57, 60, 61, 62, 63, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 83, 84, 85, 87, 88, 90, 91, 93, 95, 96, 98, 99, 100

Offset: 1

Clark Kimberling, Feb 26 2012

nonn

See the Comments at A207669.

			(See the Example section of A207669.)

Cf. A207669.

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 *)

cp's OEIS Frontend

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

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