cp's OEIS Frontend

A208135 Numbers that match polynomials over {0,1} that have a factor containing a negative coefficient.

%I A208135 #14 Aug 03 2025 06:52:36
%S A208135 9,18,21,27,33,35,36,39,42,45,49,54,57,63,65,66,70,72,75,78,84,90,93,
%T A208135 98,99,105,108,114,126,129,130,132,133,135,140,141,144,147,150,153,
%U A208135 155,156,159,161,165,168,175,177,180,183,186,189,195,196,198,201
%N A208135 Numbers that match polynomials over {0,1} that have a factor containing a negative coefficient.
%C A208135 The polynomials having coefficients in {0,1} are enumerated at A206073. They include the following:
%C A208135   p(1,x) = 1
%C A208135   p(2,x) = x
%C A208135   p(3,x) = x + 1
%C A208135   p(9,x) = x^3 + 1 = (x + 1)*(x^2 - x + 1)
%C A208135   p(18,x) = x*(x + 1)*(x^2 - x + 1)
%C A208135   p(33,x) = (x + 1)*(x^4 - x^3 + x^2 - x + 1).
%C A208135 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 A208135 The first few polynomial factors having a negative coefficient are as follows:
%e A208135   x^2 - x + 1 divides p(n,x) for n=9,18,21,27,36,42,...
%e A208135   x^4 - x^3 + x^2 - x + 1 divides p(n,x) for n=33,66,...
%e A208135   x^3 - x^2 + 1 divides p(n,x) for n=35,70,...
%e A208135   x^4 - x^3 + x^2 + 1 divides p(n,x) for n=39,...
%e A208135   x^3 - x + 1 divides p(n,x) for n=49,...
%e A208135   x^4 + x^2 - x + 1 divides p(n,x) for n=57,...
%e A208135 In A208136, the duplicates (such as 18, 21, 27, 36, 42, ...) are omitted.
%t A208135 t = Table[IntegerDigits[n, 2], {n, 1, 3000}];
%t A208135 b[n_] := Reverse[Table[x^k, {k, 0, n}]];
%t A208135 p[n_, x_] := p[n, x] = t[[n]].b[-1 + Length[t[[n]]]];
%t A208135 TableForm[Table[{n, p[n, x], Factor[p[n, x]]}, {n, 1, 250}]];
%t A208135 Map[#[[1]] &, DeleteCases[Table[{z,
%t A208135    Select[Flatten[Table[CoefficientList[#[[n]], x],
%t A208135   {n, 1, Length[#]}]] &[Factor[p[z, x]]], # < 0 &]},
%t A208135   {z, 1, 250}], {_, {}}]]
%t A208135 (* _Peter J. C. Moses_, Feb 22 2012 *)
%Y A208135 Cf. A208136, A206073, A206284.
%K A208135 nonn
%O A208135 1,1
%A A208135 _Clark Kimberling_, Feb 23 2012