cp's OEIS Frontend

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

%I A206073 #16 Mar 30 2012 18:58:12
%S A206073 10,11,101,111,1011,1101,10001,10011,10111,11001,11101,11111,100101,
%T A206073 101001,101011,101111,110101,110111,111011,111101,1000011,1000101,
%U A206073 1000111,1001001,1001101,1001111,1010001,1010011,1010111,1011001
%N A206073 Binary numbers that represent irreducible polynomials over the rationals with coefficients restricted to {0,1}.
%C A206073 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.)
%e A206073 The matching of binary numbers to the first six polynomials irreducible over the field of rational numbers:
%e A206073 10 .... x
%e A206073 11 .... x + 1
%e A206073 101 ... x^2 + 1
%e A206073 111 ... x^2 + x + 1
%e A206073 1011 .. x^3 + x + 1
%t A206073 t = Table[IntegerDigits[n, 2], {n, 1, 850}];
%t A206073 b[n_] := Reverse[Table[x^k, {k, 0, n}]]
%t A206073 p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
%t A206073 Table[p[n, x], {n, 1, 15}]
%t A206073 u = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],
%t A206073 AppendTo[u, n]], {n, 300}];
%t A206073 u                         (* A206074 *)
%t A206073 Complement[Range[200], u] (* A205783 *)
%t A206073 b[n_] := FromDigits[IntegerDigits[u, 2][[n]]]
%t A206073 Table[b[n], {n, 1, 40}]   (* A206073 *)
%Y A206073 Cf. A171000 (irreducible Boolean polynomials).
%Y A206073 Cf. A205783 (complement), A206074 (base 10).
%K A206073 nonn,base
%O A206073 1,1
%A A206073 _Clark Kimberling_, Feb 03 2012