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%I #10 Dec 04 2016 19:46:26
%S 141,177,183,237,282,354,366,427,474,555,564,573,663,669,699,708,711,
%T 717,723,732,741,753,813,849,854,871,885,909,923,933,948,951,1047,
%U 1085,1110,1115,1119,1128,1131,1145,1146,1253,1265,1299,1326,1335
%N Numbers that match polynomials with coefficients in {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(141,x) = x^7 + x^3 + + x^2 + 1 = (x + 1)*f(x), where
%C f(x) = x^6 - x^5 + x^4 - x^3 + 2*x^2 - x + 1. This shows that a factor of p(141,x) has a factor that has 2 as a coefficient. Actually, 141 is the least n for which p(n,x) has a coefficient not in {-1,0,1}.
%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 A208179 are disjoint.
%e The first five polynomial factors having 2 as a coefficient are indicated here:
%e ...
%e n ..... coefficients of a factor of p(n,x)
%e 141 ... 1, -1, 2, -1, 1, -1, 1 (see Comments)
%e 177 ... 1, -1, 1, -1, 2, -1
%e 183 ... 1, 0, 1, -1, 2, -1, 1
%e 237 ... 1, -1, 2, -1, 1, 0, 1
%e 282 ... 1, -1, 2, -1, 1, -1, 1 (same as for n=141)
%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]]}, {n, 1, 1500}]];
%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, 1500}]]]]], {_, {}}]
%t Map[#[[1]] &, %]
%t (* _Peter J. C. Moses_, Feb 22 1012 *)
%Y Cf. A208180, A206073, A206284, A208181, A208182.
%K nonn
%O 1,1
%A _Clark Kimberling_, Feb 24 2012
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