

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
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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



