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.
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 24 2012
STATUS
approved