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%I #7 Dec 04 2016 19:46:27
%S 8421,8853,9093,10311,10353,10389,10437,10563,10689,10821,10833,10839,
%T 10869,11157,12183,12453,14469,14973,14997,16779,16842,17055,17465,
%U 17706,18186,18515,18639,19985,20025,20622,20643,20706,20778
%N Numbers that match polynomials over {0,1} that have a factor containing -3 as a coefficient; see Comments.
%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(4,x) = x^2
%C p(8421,x) =1 + x^2 + x^5 + x^6 + x^7 + x^13
%C = (1 + x) (1 + x + x^2)*f(x), where
%C f(x) = 1 - 2 x + 3 x^2 - 3 x^3 + 2 x^4 - x^7 + 2 x^8 - 2 x^9 + x^10.
%C This show that a factor of p(8421,x) has a factor that has -3 as a coefficient. Actually, 8421 is the least n for which p(n,x) has a coefficient not in {-2,-1,0,1,2,3}.
%C The enumeration scheme for all nonzero polynomials with coefficients in {0,1} is introduced in Comments at A206073. The sequence A206073 itself enumerates only those polynomials that are irreducible over the ring of polynomials having integer coefficients; therefore, A206073 and A208181 are disjoint.
%t t = Table[IntegerDigits[n, 2], {n, 1, 25000}];
%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, 25000}]];
%t DeleteCases[
%t Map[{#[[1]], Cases[#[[2]], {___, -3, ___}]} &,
%t Map[{#[[1]], CoefficientList[#[[2]], x]} &,
%t Map[{#[[1]], Map[#[[1]] &, #[[2]]]} &,
%t Map[{#[[1]], Rest[FactorList[#[[2]]]]} &,
%t Table[{n, Factor[p[n, x]]}, {n, 1, 24900}]]]]], {_, {}}]
%t Map[#[[1]] &, %] (* A208182 *)
%t (* _Peter J. C. Moses_, Feb 22 1012 *)
%Y Cf. A208179, A206073, A206284, A208180, A208181.
%K nonn
%O 1,1
%A _Clark Kimberling_, Feb 24 2012
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