A208181 Numbers that match polynomials over {0,1} that have a factor containing 3 as a coefficient; see Comments.

2229, 2613, 2757, 2769, 4458, 5226, 5514, 5538, 7335, 8373, 8421, 8589, 8853, 8913, 8916, 8919, 8949, 9093, 9485, 10293, 10311, 10353, 10389, 10437, 10452, 10461, 10563, 10677, 10689, 10821, 10833, 10839, 10869, 11013, 11028, 11031

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(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}.

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..36.

Mathematica

t = Table[IntegerDigits[n, 2], {n, 1, 15000}];
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, 15000}]];
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, 14900}]]]]], {_, {}}]
Map[#[[1]] &, %]   (* A208181 *)

Crossrefs

Cf. A208179, A206073, A206284, A208180, A208182.

Sequence in context: A178919 A205225 A236496 * A252644 A205046 A205039

Adjacent sequences: A208178 A208179 A208180 * A208182 A208183 A208184

Keyword

nonn

Author

Clark Kimberling, Feb 24 2012

Status

approved