A208179 Numbers that match polynomials with coefficients in {0,1} that have a factor containing 2 as a coefficient; see Comments.

141, 177, 183, 237, 282, 354, 366, 427, 474, 555, 564, 573, 663, 669, 699, 708, 711, 717, 723, 732, 741, 753, 813, 849, 854, 871, 885, 909, 923, 933, 948, 951, 1047, 1085, 1110, 1115, 1119, 1128, 1131, 1145, 1146, 1253, 1265, 1299, 1326, 1335

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(141,x) = x^7 + x^3 + + x^2 + 1 = (x + 1)*f(x), where

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}.

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.

Links

Table of n, a(n) for n=1..46.

Example

The first five polynomial factors having 2 as a coefficient are indicated here:
...
n ..... coefficients of a factor of p(n,x)
141 ... 1, -1, 2, -1, 1, -1, 1  (see Comments)
177 ... 1, -1, 1, -1, 2, -1
183 ... 1, 0, 1, -1, 2, -1, 1
237 ... 1, -1, 2, -1, 1, 0, 1
282 ... 1, -1, 2, -1, 1, -1, 1  (same as for n=141)

Mathematica

t = Table[IntegerDigits[n, 2], {n, 1, 3000}];
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, 1500}]];
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, 1500}]]]]], {_, {}}]
Map[#[[1]] &, %]
(* Peter J. C. Moses, Feb 22 1012 *)

Crossrefs

Cf. A208180, A206073, A206284, A208181, A208182.

Sequence in context: A235689 A045935 A144799 * A368670 A045936 A228598

Adjacent sequences: A208176 A208177 A208178 * A208180 A208181 A208182

Keyword

nonn

Author

Clark Kimberling, Feb 24 2012

Status

approved