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A208136 Subsequence of A208135 with numbers that match duplicate factors deleted. 2

%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

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Last modified September 21 10:03 EDT 2020. Contains 337268 sequences. (Running on oeis4.)