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.

Previous Showing 11-13 of 13 results.

A207968 Ternary numbers that match irreducible polynomials over {0,1,2}.

Original entry on oeis.org

10, 11, 12, 20, 21, 22, 101, 102, 111, 112, 122, 201, 202, 211, 212, 221, 222, 1002, 1011, 1021, 1022, 1101, 1102, 1112, 1121, 1201, 1202, 1211, 1222, 2001, 2011, 2012, 2021, 2022, 2102, 2111, 2122, 2201, 2202, 2212, 2221, 10001, 10002, 10011
Offset: 1

Views

Author

Clark Kimberling, Feb 21 2012

Keywords

Comments

These are base 3 representations for the numbers (in base 10) at A207966.

Crossrefs

Cf. A207966, A207966, A206073 (base 2).

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 3], {n, 1, 850}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
Table[p[n, x], {n, 1, 15}]
u = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],
  AppendTo[u, n]], {n, 300}]; u  (* A207966 *)
Complement[Range[200], u]        (* A207967 *)
b[n_] := FromDigits[IntegerDigits[u, 3][[n]]]
Table[b[n], {n, 1, 50}]          (* A207968 *)

A208135 Numbers that match polynomials over {0,1} that have a factor containing a negative coefficient.

Original entry on oeis.org

9, 18, 21, 27, 33, 35, 36, 39, 42, 45, 49, 54, 57, 63, 65, 66, 70, 72, 75, 78, 84, 90, 93, 98, 99, 105, 108, 114, 126, 129, 130, 132, 133, 135, 140, 141, 144, 147, 150, 153, 155, 156, 159, 161, 165, 168, 175, 177, 180, 183, 186, 189, 195, 196, 198, 201
Offset: 1

Views

Author

Clark Kimberling, Feb 23 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(9,x) = x^3 + 1 = (x + 1)*(x^2 - x + 1)
p(18,x) = x*(x + 1)*(x^2 - x + 1)
p(33,x) = (x + 1)*(x^4 - x^3 + x^2 - x + 1).
A208135 gives those n for which p(n,x) has a factor containing a negative coefficient; A208136 is a subsequence of A208135 in which, for each p(n,x), there is a factor containing a negative coefficient, and that factor has not already occurred for some p(k,x) with k

Examples

			The first few polynomial factors having a negative coefficient are as follows:
  x^2 - x + 1 divides p(n,x) for n=9,18,21,27,36,42,...
  x^4 - x^3 + x^2 - x + 1 divides p(n,x) for n=33,66,...
  x^3 - x^2 + 1 divides p(n,x) for n=35,70,...
  x^4 - x^3 + x^2 + 1 divides p(n,x) for n=39,...
  x^3 - x + 1 divides p(n,x) for n=49,...
  x^4 + x^2 - x + 1 divides p(n,x) for n=57,...
In A208136, the duplicates (such as 18, 21, 27, 36, 42, ...) are omitted.

Crossrefs

Cf. A208136, A206073, A206284.

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, 250}]];
Map[#[[1]] &, DeleteCases[Table[{z,
   Select[Flatten[Table[CoefficientList[#[[n]], x],
  {n, 1, Length[#]}]] &[Factor[p[z, x]]], # < 0 &]},
  {z, 1, 250}], {_, {}}]]
(* Peter J. C. Moses, Feb 22 2012 *)

A208136 Subsequence of A208135 with numbers that match duplicate factors deleted.

Original entry on oeis.org

9, 33, 35, 39, 49, 57, 65, 129, 133, 135, 147, 159, 161, 183, 201, 215, 225, 235, 237, 249, 259, 267, 287, 291, 303, 371, 385, 393, 413, 417, 423, 427, 459, 489, 497, 519, 525, 527, 537, 543, 573, 579, 591, 605, 609, 615, 633, 651
Offset: 1

Views

Author

Clark Kimberling, Feb 23 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(9,x) = x^3 + 1 = (x + 1)*(x^2 - x + 1)
p(18,x) = x*(x + 1)*(x^2 - x + 1)
p(33,x) = (x + 1)*(x^4 - x^3 + x^2 - x + 1).
A208135 gives those n for which p(n,x) has a factor containing a negative coefficient; A208136 is a subsequence of A208135 in which, for each p(n,x), there is a factor containing a negative coefficient, and that factor has not already occurred for some p(k,x) with k

Examples

			The first few polynomial factors having a negative coefficient are as follows:
  x^2 - x + 1 divides p(n,x) for n=9,18,21,27,36,42,...
  x^4 - x^3 + x^2 - x + 1 divides p(n,x) for n=33,66,...
  x^3 - x^2 + 1 divides p(n,x) for n=35,70,...
  x^4 - x^3 + x^2 + 1 divides p(n,x) for n=39,...
  x^3 - x + 1 divides p(n,x) for n=49,...
  x^4 + x^2 - x + 1 divides p(n,x) for n=57,...
In A208136, the duplicates (such as 18, 21, 27, 36, 42, ...) are omitted.

Crossrefs

Cf. A208135, A206073, A206284.

Programs

  • Mathematica
    Remove["Global`*"];
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, 900}]];
ans = DeleteCases[Table[{z, Cases[Sign[
       Table[CoefficientList[#[[n]], x], {n, 1, Length[#]}] &[Factor[p[z, x]]]], {_, -1, _}]}, {z, 1, 700}], {_, {}}];
n = 1; While[Length[ans] >= n,
ans = Delete[ans, Map[Take[{#[[1]]}] &, Rest[Position[ans, Flatten[ans[[n]][[2]]]]]]]; n++];
Map[#[[1]] &, ans]
(* Peter J. C. Moses, Feb 22 2012 *)
Previous Showing 11-13 of 13 results.

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