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A208181 Numbers that match polynomials over {0,1} that have a factor containing 3 as a coefficient; see Comments. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 7 15:24 EST 2021. Contains 349581 sequences. (Running on oeis4.)