A206073 Binary numbers that represent irreducible polynomials over the rationals with coefficients restricted to {0,1}.

10, 11, 101, 111, 1011, 1101, 10001, 10011, 10111, 11001, 11101, 11111, 100101, 101001, 101011, 101111, 110101, 110111, 111011, 111101, 1000011, 1000101, 1000111, 1001001, 1001101, 1001111, 1010001, 1010011, 1010111, 1011001

Offset: 1

Author

Clark Kimberling, Feb 03 2012

Keywords

• nonn
• base

Comments

The polynomial x^d(0) + x^d(1) + ... + d(n), where d(i) is 0 or 1 for 0<=i<=n and d(0)=1, matches the binary number d(0)d(1)...d(n). (This is an enumeration of all the nonzero polynomials with coefficients in {0,1}, not just those that are irreducible.)

Examples

			The matching of binary numbers to the first six polynomials irreducible over the field of rational numbers:
10 .... x
11 .... x + 1
101 ... x^2 + 1
111 ... x^2 + x + 1
1011 .. x^3 + x + 1
		

Crossrefs

Cf. A171000 (irreducible Boolean polynomials).

Cf. A205783 (complement), A206074 (base 10).

Programs

Mathematica

t = Table[IntegerDigits[n, 2], {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                         (* A206074 *)
Complement[Range[200], u] (* A205783 *)
b[n_] := FromDigits[IntegerDigits[u, 2][[n]]]
Table[b[n], {n, 1, 40}]   (* A206073 *)