A208135 - OEIS

%I #10 Dec 04 2016 19:46:26

%S 9,18,21,27,33,35,36,39,42,45,49,54,57,63,65,66,70,72,75,78,84,90,93,

%T 98,99,105,108,114,126,129,130,132,133,135,140,141,144,147,150,153,

%U 155,156,159,161,165,168,175,177,180,183,186,189,195,196,198,201

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

%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 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]]}, {n, 1, 250}]];

%t Map[#[[1]] &, DeleteCases[Table[{z,

%t    Select[Flatten[Table[CoefficientList[#[[n]], x],

%t   {n, 1, Length[#]}]] &[Factor[p[z, x]]], # < 0 &]},

%t   {z, 1, 250}], {_, {}}]]

%t (* _Peter J. C. Moses_, Feb 22 1012 *)

%Y Cf. A208136, A206073, A206284.

%K nonn

%O 1,1

%A _Clark Kimberling_, Feb 23 2012