A207966 Numbers that match irreducible polynomials over {0,1,2}.

3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 17, 19, 20, 22, 23, 25, 26, 29, 31, 34, 35, 37, 38, 41, 43, 46, 47, 49, 53, 55, 58, 59, 61, 62, 65, 67, 71, 73, 74, 77, 79, 82, 83, 85, 86, 89, 92, 94, 95, 97, 98, 101, 103, 106, 107, 109, 110, 113, 115, 118, 121, 122, 125, 127

Offset: 1

Author

Clark Kimberling, Feb 21 2012

Keywords

• nonn

Comments

Each n > 1 matches a polynomial having coefficients in {0,1,2}, determined by the prime factorization of n.

Write n = p(1)^e(1) * p(2)^e(2) * ... * p(k)^e(k).

The matching polynomial is then

p(n,x) = e(1) + e(2)x + e(3)x^2 + ... + e(k)x^k.

Identities:

p(m*n) = p(m,x) + p(n,x);

p(m*n) = p(gcd(m,n),x) + p(lcm(m,n),x).

For an analogous enumeration of polynomials over {0,1}, see A206284.

"Irreducible" refers to irreducibility over the field of rational numbers.

Examples

			Polynomials having coefficients in {0,1,2} are
matched to the positive integers as follows:
n ... p[n,x] .. irreducible
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 .. 1+x^2 ... yes
		

Crossrefs

Cf. A207967, A207968, A206284.

Programs

Mathematica

t = Table[IntegerDigits[n, 3], {n, 1, 850}];
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]],
  AppendTo[u, n]], {n, 300}]; u  (* A207966 *)
Complement[Range[200], u]        (* A207967 *)
b[n_] := FromDigits[IntegerDigits[u, 3][[n]]]
Table[b[n], {n, 1, 50}]          (* A207968 *)