A208136 Subsequence of A208135 with numbers that match duplicate factors deleted.

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

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