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

1,1

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<n.

Links

Table of n, a(n) for n=1..56.

Example

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.

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 *)

Crossrefs

Cf. A208136, A206073, A206284.

Sequence in context: A015785 A366290 A316438 * A306382 A373351 A222623

Adjacent sequences: A208132 A208133 A208134 * A208136 A208137 A208138

Keyword

nonn

Author

Clark Kimberling, Feb 23 2012

Status

approved