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A208182 Numbers that match polynomials over {0,1} that have a factor containing -3 as a coefficient; see Comments. 4
8421, 8853, 9093, 10311, 10353, 10389, 10437, 10563, 10689, 10821, 10833, 10839, 10869, 11157, 12183, 12453, 14469, 14973, 14997, 16779, 16842, 17055, 17465, 17706, 18186, 18515, 18639, 19985, 20025, 20622, 20643, 20706, 20778 (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(8421,x) =1 + x^2 + x^5 + x^6 + x^7 + x^13

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

f(x) =  1 - 2 x + 3 x^2 - 3 x^3 + 2 x^4 - x^7 + 2 x^8 - 2 x^9 + x^10.

This show that a factor of p(8421,x) has a factor that has -3 as a coefficient.  Actually, 8421 is the least n for which p(n,x) has a coefficient not in {-2,-1,0,1,2,3}.

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..33.

MATHEMATICA

t = Table[IntegerDigits[n, 2], {n, 1, 25000}];

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, 25000}]];

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, 24900}]]]]], {_, {}}]

Map[#[[1]] &, %]   (* A208182 *)

(* Peter J. C. Moses, Feb 22 1012 *)

CROSSREFS

Cf. A208179, A206073, A206284, A208180, A208181.

Sequence in context: A232300 A214117 A237137 * A093221 A023321 A116258

Adjacent sequences:  A208179 A208180 A208181 * A208183 A208184 A208185

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 24 2012

STATUS

approved

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Last modified October 5 23:01 EDT 2022. Contains 357261 sequences. (Running on oeis4.)