%I
%S 9,33,35,39,49,57,65,129,133,135,147,159,161,183,201,215,225,235,237,
%T 249,259,267,287,291,303,371,385,393,413,417,423,427,459,489,497,519,
%U 525,527,537,543,573,579,591,605,609,615,633,651
%N Subsequence of A208135 with numbers that match duplicate factors deleted.
%C The polynomials having coefficients in {0,1} are enumerated at A206073. They include the following:
%C p(1,x) = 1
%C p(2,x) = x
%C p(3,x) = x + 1
%C p(9,x) = x^3 + 1 = (x + 1)(x^2  x + 1)
%C p(18,x) = x(x + 1)(x^2  x + 1)
%C p(33,x) = (x + 1)(x^4  x^3 + x^2  x + 1).
%C 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.
%e The first few polynomial factors having a negative
%e coefficients are as follows:
%e x^2  x + 1 divides p(n,x) for n=9,18,21,27,36,42,...
%e x^4  x^3 + x^2  x + 1 divides p(n,x) for n=33,66,...
%e x^3  x^2 + 1 divides p(n,x) for n=35,70,...
%e x^4  x^3 + x^2 + 1 divides p(n,x) for n=39,...
%e x^3  x + 1 divides p(n,x) for n=49,...
%e x^4 + x^2  x + 1 divides p(n,x) for n=57,...
%e In A208136, the duplicates (such as 18, 21, 27, 36,
%e 42, ...) are omitted.
%t Remove["Global`*"];
%t t = Table[IntegerDigits[n, 2], {n, 1, 3000}];
%t b[n_] := Reverse[Table[x^k, {k, 0, n}]];
%t p[n_, x_] := p[n, x] = t[[n]].b[1 + Length[t[[n]]]];
%t TableForm[Table[{n, p[n, x], Factor[p[n, x]]},
%t {n, 1, 900}]];
%t ans = DeleteCases[Table[{z, Cases[Sign[
%t Table[CoefficientList[#[[n]], x], {n, 1, Length[#]}] &[Factor[p[z, x]]]], {___, 1, ___}]}, {z, 1, 700}], {_, {}}];
%t n = 1; While[Length[ans] >= n,
%t ans = Delete[ans, Map[Take[{#[[1]]}] &,
%t Rest[Position[ans, Flatten[ans[[n]][[2]]]]]]]; n++];
%t Map[#[[1]] &, ans]
%t (* _Peter J. C. Moses_, Feb 22 1012 *)
%Y Cf. A208135, A206073, A206284.
%K nonn
%O 1,1
%A _Clark Kimberling_, Feb 23 2012
