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A208135 Numbers that match polynomials over {0,1} that have a factor containing a negative coefficient. 2
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 (list; graph; refs; listen; history; text; internal format)
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

coefficients 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 1012 *)

CROSSREFS

Cf. A208136, A206073, A206284.

Sequence in context: A254066 A015785 A316438 * A306382 A222623 A141469

Adjacent sequences:  A208132 A208133 A208134 * A208136 A208137 A208138

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 23 2012

STATUS

approved

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Last modified September 25 06:32 EDT 2020. Contains 337335 sequences. (Running on oeis4.)