A208135 Numbers that match polynomials over {0,1} that have a factor containing a negative coefficient.

9, 18, 21, 27, 33, 35, 36, 39, 42, 45, 49, 54, 57, 63, 65, 66, 70, 72, 75, 78, 84, 90, 93, 98, 99, 105, 108, 114, 126, 129, 130, 132, 133, 135, 140, 141, 144, 147, 150, 153, 155, 156, 159, 161, 165, 168, 175, 177, 180, 183, 186, 189, 195, 196, 198, 201

Offset: 1

Author

Clark Kimberling, Feb 23 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(9,x) = x^3 + 1 = (x + 1)*(x^2 - x + 1)

p(18,x) = x*(x + 1)*(x^2 - x + 1)

p(33,x) = (x + 1)*(x^4 - x^3 + x^2 - x + 1).

A208135 gives those n for which p(n,x) has a factor containing a negative coefficient; A208136 is a subsequence of A208135 in which, for each p(n,x), there is a factor containing a negative coefficient, and that factor has not already occurred for some p(k,x) with k

Examples

			The first few polynomial factors having a negative coefficient are as follows:
  x^2 - x + 1 divides p(n,x) for n=9,18,21,27,36,42,...
  x^4 - x^3 + x^2 - x + 1 divides p(n,x) for n=33,66,...
  x^3 - x^2 + 1 divides p(n,x) for n=35,70,...
  x^4 - x^3 + x^2 + 1 divides p(n,x) for n=39,...
  x^3 - x + 1 divides p(n,x) for n=49,...
  x^4 + x^2 - x + 1 divides p(n,x) for n=57,...
In A208136, the duplicates (such as 18, 21, 27, 36, 42, ...) are omitted.
		

Crossrefs

Cf. A208136, A206073, A206284.

Programs

Mathematica

t = Table[IntegerDigits[n, 2], {n, 1, 3000}];
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, 250}]];
Map[#[[1]] &, DeleteCases[Table[{z,
   Select[Flatten[Table[CoefficientList[#[[n]], x],
  {n, 1, Length[#]}]] &[Factor[p[z, x]]], # < 0 &]},
  {z, 1, 250}], {_, {}}]]
(* Peter J. C. Moses, Feb 22 2012 *)

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