A208182 Numbers that match polynomials over {0,1} that have a factor containing -3 as a coefficient; see Comments.

8421, 8853, 9093, 10311, 10353, 10389, 10437, 10563, 10689, 10821, 10833, 10839, 10869, 11157, 12183, 12453, 14469, 14973, 14997, 16779, 16842, 17055, 17465, 17706, 18186, 18515, 18639, 19985, 20025, 20622, 20643, 20706, 20778

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(8421,x) =1 + x^2 + x^5 + x^6 + x^7 + x^13

= (1 + x) (1 + x + x^2)*f(x), where

f(x) = 1 - 2 x + 3 x^2 - 3 x^3 + 2 x^4 - x^7 + 2 x^8 - 2 x^9 + x^10.

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}.

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.

Links

Table of n, a(n) for n=1..33.

Mathematica

t = Table[IntegerDigits[n, 2], {n, 1, 25000}];
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, 25000}]];
DeleteCases[
Map[{#[[1]], Cases[#[[2]], {___, -3, ___}]} &,
  Map[{#[[1]], CoefficientList[#[[2]], x]} &,
   Map[{#[[1]], Map[#[[1]] &, #[[2]]]} &,
    Map[{#[[1]], Rest[FactorList[#[[2]]]]} &,
     Table[{n, Factor[p[n, x]]}, {n, 1, 24900}]]]]], {_, {}}]
Map[#[[1]] &, %]   (* A208182 *)
(* Peter J. C. Moses, Feb 22 1012 *)

Crossrefs

Cf. A208179, A206073, A206284, A208180, A208181.

Sequence in context: A232300 A214117 A237137 * A093221 A023321 A116258

Adjacent sequences: A208179 A208180 A208181 * A208183 A208184 A208185

Keyword

nonn

Author

Clark Kimberling, Feb 24 2012

Status

approved