%I #8 Dec 04 2016 19:46:27

%S 663,669,741,933,1326,1338,1421,1482,1866,2163,2181,2199,2229,2247,

%T 2289,2387,2469,2499,2577,2589,2613,2631,2643,2649,2652,2661,2676,

%U 2679,2757,2769,2842,2949,2964,2973,3115,3129,3237,3241,3297,3395

%N Numbers that match polynomials over {0,1} that have a factor containing -2 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(663,x) = 1 + x + x^2 + x^4 + x^7 + x^9 = (x + 1)*f(x), where f(x) = 1 + x^2 - x^3 + 2 x^4 - 2 x^5 + 2 x^6 - x^7 + x^8. This show that a factor of p(663,x) has a factor that has -2 as a coefficient. Actually, 663 is the least n for which p(n,x) has a coefficient not in {-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 A208180 are disjoint.

%t t = Table[IntegerDigits[n, 2], {n, 1, 4000}];

%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, 4000}]];

%t DeleteCases[

%t Map[{#[[1]], Cases[#[[2]], {___, -2, ___}]} &,

%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, 3600}]]]]], {_, {}}]

%t Map[#[[1]] &, %] (* A208180 *)

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

%Y Cf. A208179, A206073, A206284, A208181, A208182.

%K nonn

%O 1,1

%A _Clark Kimberling_, Feb 24 2012