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%I #7 Mar 30 2012 18:58:13
%S 2229,2613,2757,2769,4458,5226,5514,5538,7335,8373,8421,8589,8853,
%T 8913,8916,8919,8949,9093,9485,10293,10311,10353,10389,10437,10452,
%U 10461,10563,10677,10689,10821,10833,10839,10869,11013,11028,11031
%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(2229,x) =1 + x^2 + x^4 + x^5 + x^7 + x^11= (1+x)*f(x), where f(x) = 1 - x + 2 x^2 - 2 x^3 + 3 x^4 - 2 x^5 + 2 x^6 - x^7 + x^8 - x^9 + x^10. This show that a factor of p(2229,x) has a factor that has 3 as a coefficient. Actually, 2229 is the least n for which p(n,x) has a coefficient not in {-2,-1,0,1,2}.
%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, 15000}];
%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, 15000}]];
%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, 14900}]]]]], {_, {}}]
%t Map[#[[1]] &, %] (* A208181 *)
%Y Cf. A208179, A206073, A206284, A208180, A208182.
%K nonn
%O 1,1
%A _Clark Kimberling_, Feb 24 2012
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