A208180 Numbers that match polynomials over {0,1} that have a factor containing -2 as a coefficient; see Comments.

663, 669, 741, 933, 1326, 1338, 1421, 1482, 1866, 2163, 2181, 2199, 2229, 2247, 2289, 2387, 2469, 2499, 2577, 2589, 2613, 2631, 2643, 2649, 2652, 2661, 2676, 2679, 2757, 2769, 2842, 2949, 2964, 2973, 3115, 3129, 3237, 3241, 3297, 3395

Offset: 1

Author

Clark Kimberling, Feb 24 2012

Keywords

• nonn

Comments

The polynomials having coefficients in {0,1} are enumerated at A206073. They include the following:

p(1,x) = 1

p(2,x) = x

p(3,x) = x + 1

p(4,x) = x^2

p(663,x) = 1 + x + x^2 + x^4 + x^7 + x^9 = (x + 1)*f(x), where f(x) = 1 + x^2 - x^3 + 2 x^4 - 2 x^5 + 2 x^6 - x^7 + x^8. This show that a factor of p(663,x) has a factor that has -2 as a coefficient. Actually, 663 is the least n for which p(n,x) has a coefficient not in {-1,0,1,2}.

The enumeration scheme for all nonzero polynomials with coefficients in {0,1} is introduced in Comments at A206073. The sequence A206073 itself enumerates only those polynomials that are irreducible over the ring of polynomials having integer coefficients; therefore, A206073 and A208180 are disjoint.

Crossrefs

Cf. A208179, A206073, A206284, A208181, A208182.

Programs

Mathematica

t = Table[IntegerDigits[n, 2], {n, 1, 4000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := p[n, x] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n, x], Factor[p[n, x]]}, {n, 1, 4000}]];
DeleteCases[
Map[{#[[1]], Cases[#[[2]], {_, -2, _}]} &,
  Map[{#[[1]], CoefficientList[#[[2]], x]} &,
   Map[{#[[1]], Map[#[[1]] &, #[[2]]]} &,
    Map[{#[[1]], Rest[FactorList[#[[2]]]]} &,
     Table[{n, Factor[p[n, x]]}, {n, 1, 3600}]]]]], {_, {}}]
Map[#[[1]] &, %]   (* A208180 *)
(* Peter J. C. Moses, Feb 22 2012 *)