

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
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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..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



