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A208180
Numbers that match polynomials over {0,1} that have a factor containing -2 as a coefficient; see Comments.
4
663, 669, 741, 933, 1326, 1338, 1421, 1482, 1866, 2163, 2181, 2199, 2229, 2247, 2289, 2387, 2469, 2499, 2577, 2589, 2613, 2631, 2643, 2649, 2652, 2661, 2676, 2679, 2757, 2769, 2842, 2949, 2964, 2973, 3115, 3129, 3237, 3241, 3297, 3395
OFFSET
1,1
COMMENTS
The polynomials having coefficients in {0,1} are enumerated at A206073. They include the following:
p(1,x) = 1
p(2,x) = x
p(3,x) = x + 1
p(4,x) = x^2
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}.
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.
MATHEMATICA
t = Table[IntegerDigits[n, 2], {n, 1, 4000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := p[n, x] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n, x], Factor[p[n, x]]}, {n, 1, 4000}]];
DeleteCases[
Map[{#[[1]], Cases[#[[2]], {___, -2, ___}]} &,
Map[{#[[1]], CoefficientList[#[[2]], x]} &,
Map[{#[[1]], Map[#[[1]] &, #[[2]]]} &,
Map[{#[[1]], Rest[FactorList[#[[2]]]]} &,
Table[{n, Factor[p[n, x]]}, {n, 1, 3600}]]]]], {_, {}}]
Map[#[[1]] &, %] (* A208180 *)
(* Peter J. C. Moses, Feb 22 1012 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 24 2012
STATUS
approved