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A208136 Subsequence of A208135 with numbers that match duplicate factors deleted. 2
9, 33, 35, 39, 49, 57, 65, 129, 133, 135, 147, 159, 161, 183, 201, 215, 225, 235, 237, 249, 259, 267, 287, 291, 303, 371, 385, 393, 413, 417, 423, 427, 459, 489, 497, 519, 525, 527, 537, 543, 573, 579, 591, 605, 609, 615, 633, 651 (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..48.

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

Remove["Global`*"];

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, 900}]];

ans = DeleteCases[Table[{z, Cases[Sign[

       Table[CoefficientList[#[[n]], x], {n, 1, Length[#]}] &[Factor[p[z, x]]]], {___, -1, ___}]}, {z, 1, 700}], {_, {}}];

n = 1; While[Length[ans] >= n,

ans = Delete[ans, Map[Take[{#[[1]]}] &,

    Rest[Position[ans, Flatten[ans[[n]][[2]]]]]]]; n++];

Map[#[[1]] &, ans]

(* Peter J. C. Moses, Feb 22 1012 *)

CROSSREFS

Cf. A208135, A206073, A206284.

Sequence in context: A061913 A264512 A231764 * A130444 A177697 A287534

Adjacent sequences:  A208133 A208134 A208135 * A208137 A208138 A208139

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 23 2012

STATUS

approved

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Last modified August 14 03:05 EDT 2020. Contains 336477 sequences. (Running on oeis4.)