cp's OEIS Frontend

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

%I A208136 #14 Aug 03 2025 16:08:50
%S A208136 9,33,35,39,49,57,65,129,133,135,147,159,161,183,201,215,225,235,237,
%T A208136 249,259,267,287,291,303,371,385,393,413,417,423,427,459,489,497,519,
%U A208136 525,527,537,543,573,579,591,605,609,615,633,651
%N A208136 Subsequence of A208135 with numbers that match duplicate factors deleted.
%C A208136 The polynomials having coefficients in {0,1} are enumerated at A206073. They include the following:
%C A208136   p(1,x) = 1
%C A208136   p(2,x) = x
%C A208136   p(3,x) = x + 1
%C A208136   p(9,x) = x^3 + 1 = (x + 1)*(x^2 - x + 1)
%C A208136   p(18,x) = x*(x + 1)*(x^2 - x + 1)
%C A208136   p(33,x) = (x + 1)*(x^4 - x^3 + x^2 - x + 1).
%C A208136 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 A208136 The first few polynomial factors having a negative coefficient are as follows:
%e A208136   x^2 - x + 1 divides p(n,x) for n=9,18,21,27,36,42,...
%e A208136   x^4 - x^3 + x^2 - x + 1 divides p(n,x) for n=33,66,...
%e A208136   x^3 - x^2 + 1 divides p(n,x) for n=35,70,...
%e A208136   x^4 - x^3 + x^2 + 1 divides p(n,x) for n=39,...
%e A208136   x^3 - x + 1 divides p(n,x) for n=49,...
%e A208136   x^4 + x^2 - x + 1 divides p(n,x) for n=57,...
%e A208136 In A208136, the duplicates (such as 18, 21, 27, 36, 42, ...) are omitted.
%t A208136 Remove["Global`*"];
%t A208136 t = Table[IntegerDigits[n, 2], {n, 1, 3000}];
%t A208136 b[n_] := Reverse[Table[x^k, {k, 0, n}]];
%t A208136 p[n_, x_] := p[n, x] = t[[n]].b[-1 + Length[t[[n]]]];
%t A208136 TableForm[Table[{n, p[n, x], Factor[p[n, x]]}, {n, 1, 900}]];
%t A208136 ans = DeleteCases[Table[{z, Cases[Sign[
%t A208136        Table[CoefficientList[#[[n]], x], {n, 1, Length[#]}] &[Factor[p[z, x]]]], {___, -1, ___}]}, {z, 1, 700}], {_, {}}];
%t A208136 n = 1; While[Length[ans] >= n,
%t A208136 ans = Delete[ans, Map[Take[{#[[1]]}] &, Rest[Position[ans, Flatten[ans[[n]][[2]]]]]]]; n++];
%t A208136 Map[#[[1]] &, ans]
%t A208136 (* _Peter J. C. Moses_, Feb 22 2012 *)
%Y A208136 Cf. A208135, A206073, A206284.
%K A208136 nonn
%O A208136 1,1
%A A208136 _Clark Kimberling_, Feb 23 2012