cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 24 2012

Keywords

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 *)

A208180 Numbers that match polynomials over {0,1} that have a factor containing -2 as a coefficient; see Comments.

Original entry on oeis.org

663, 669, 741, 933, 1326, 1338, 1421, 1482, 1866, 2163, 2181, 2199, 2229, 2247, 2289, 2387, 2469, 2499, 2577, 2589, 2613, 2631, 2643, 2649, 2652, 2661, 2676, 2679, 2757, 2769, 2842, 2949, 2964, 2973, 3115, 3129, 3237, 3241, 3297, 3395
Offset: 1

Views

Author

Clark Kimberling, Feb 24 2012

Keywords

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(663,x) = 1 + x + x^2 + x^4 + x^7 + x^9 = (x + 1)*f(x), where f(x) = 1 + x^2 - x^3 + 2 x^4 - 2 x^5 + 2 x^6 - x^7 + x^8. This show that a factor of p(663,x) has a factor that has -2 as a coefficient. Actually, 663 is the least n for which p(n,x) has a coefficient not in {-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 A208180 are disjoint.

Crossrefs

Cf. A208179, A206073, A206284, A208181, A208182.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 4000}];
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, 4000}]];
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, 3600}]]]]], {_, {}}]
Map[#[[1]] &, %]   (* A208180 *)
(* Peter J. C. Moses, Feb 22 2012 *)

A208182 Numbers that match polynomials over {0,1} that have a factor containing -3 as a coefficient; see Comments.

Original entry on oeis.org

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
Offset: 1

Views

Author

Clark Kimberling, Feb 24 2012

Keywords

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.

Crossrefs

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

Programs

  • 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 2012 *)
Showing 1-3 of 3 results.

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