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

Author

Clark Kimberling, Feb 24 2012

Keywords

• nonn

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.

Examples

			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)
		

Crossrefs

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

Programs

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 2012 *)