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 1-4 of 4 results.

A206821 Numbers that match irreducible polynomials over {-1,0,1} with leading coefficient 1.

Original entry on oeis.org

2, 3, 7, 8, 10, 14, 16, 18, 21, 23, 29, 31, 35, 41, 42, 44, 48, 50, 54, 56, 60, 62, 66, 70, 72, 76, 78, 80, 82, 84, 86, 88, 93, 97, 99, 103, 109, 111, 115, 117, 123, 125, 129, 131, 137, 141, 143, 147, 153, 155, 159, 161, 165, 167, 171, 173, 179, 183, 186, 188
Offset: 1

Views

Author

Clark Kimberling, Feb 12 2012

Keywords

Comments

The monic polynomials y(n,x) having coefficients in {-1,0,1} are matched to the set N of positive integers as follows. First, the monic polynomials p(n,x) having coefficients in {0,1} are matched to N as in A206074; i.e., the polynomial x^d(0) + x^d(1) + ... + d(n), where d(i) is 0 or 1 for 0<=i<=n and d(0)=1, matches the binary number d(0)d(1)...d(n). Then monic polynomials having at least one negative coefficient are then inserted among the polynomials p(n,x), as follows: x-1 goes between x and x+1, and for k>1, the polynomials x^k-p(n,x), for 0
n ..... y(n,x) ... irreducible
1 ..... 1 ........ no
2 ..... x ........ yes
3 ..... 1+x ...... yes
4 ..... x^2 ...... no
5 .... -1+x^2 .... no
6 .... -x+x^2 .... no
7 .... -1-x+x^2 .. yes
8 ..... 1+x^2 .... yes
9 ..... x+x^2 .... no
10 .... 1+x+x^2 .. yes
11 .... x^3 ...... no
...
Guide to sequences based on the polynomials y(n,x):
A206822, irreducible
A206829, number of distinct factors
A207187, multiples of x+1
A207188, multiples of x
A207189, multiples of x-1
A207190, multiples of x^2+1
A207191, even: y(n,-x)=y(n,x)
A207192, odd: y(n,-x)=-y(n,x)

Crossrefs

Cf. A206073, A206284, A206822.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 1000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]];
p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]];
TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]];
q2[n_] := p[n - f[k] + 2];
y1 = Table[p[n], {n, 1, 4}];
Do[AppendTo[y1,Join[Table[q1[n], {n, f[k], g[k] - 1}],
   Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 8}]
y = Flatten[y1]; (* polynomials over {-1,0,1} *)
w = {}; Do[n++; If[IrreduciblePolynomialQ[y[[n]]], AppendTo[w, n]], {n, 200}]
w                          (* A206821 *)
Complement[Range[200], w]  (* A206822 *)

A207187 Numbers matching polynomials y(k,x) that have x+1 as a factor; see Comments.

Original entry on oeis.org

3, 5, 9, 13, 19, 22, 25, 27, 30, 33, 39, 43, 49, 52, 55, 59, 65, 68, 71, 83, 89, 92, 95, 101, 104, 107, 110, 113, 116, 119, 121, 124, 127, 133, 136, 139, 142, 145, 148, 151, 157, 169, 172, 175, 181, 185, 191, 194, 197, 209, 215, 218, 221, 224, 227, 230
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2012

Keywords

Comments

The polynomials y(k,x) range through all monic polynomials with coefficients in {-1,0,1}, ordered as at A206821.

Examples

			The first 13 polynomials:
1 .... 1
2 .... x
3 .... x + 1
4 .... x^2
5 .... x^2 - 1
6 .... x^2 - x
7 .... x^2 - x - 1
8 .... x^2 + 1
9 .... x^2 + x
10 ... x^2 + x + 1
11 ... x^3
12 ... x^3 - 1
13 ... x^3 - x
The list exemplifies these sequences:
A207187 (multiples of x + 1): 3,5,9,13,...
A207188 (multiples of x): 2,4,6,9,11,13,...
A207189 (multiples of x - 1): 5,6,12,13,...
A207190 (multiples of x^2 + 1): 8,20,25,27,...

Crossrefs

Cf. A206821.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 2000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]]
q2[n_] := p[n - f[k] + 2]
y1 = Table[p[n], {n, 1, 4}];
Do[AppendTo[y1,
  Join[Table[q1[n], {n, f[k], g[k] - 1}],
   Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 10}]
y = Flatten[y1]; (* monic polynomials over {-1,0,1} *)
TableForm[Table[{n, y[[n]], Factor[y[[n]]]}, {n, 1, 10}]]
Table[y[[n]] /. x -> -1, {n, 1, 300}];
Flatten[Position[%, 0]]  (* A207187 *)
Table[y[[n]] /. x -> 0, {n, 1, 300}] ;
Flatten[Position[%, 0]]  (* A207188 *)
Table[y[[n]] /. x -> 1, {n, 1, 1200}] ;
Flatten[Position[%, 0]]  (* A207189 *)
Table[y[[n]] /. x -> I, {n, 1, 600}] ;
Flatten[Position[%, 0]]  (* A207190 *)

A207188 Numbers matching polynomials y(k,x) that have x as a factor; see Comments.

Original entry on oeis.org

2, 4, 6, 9, 11, 13, 15, 17, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2012

Keywords

Comments

The polynomials y(k,x) range through all monic polynomials with coefficients in {-1,0,1}, ordered as at A206821.

Examples

			The first 13 polynomials:
1 .... 1
2 .... x
3 .... x + 1
4 .... x^2
5 .... x^2 - 1
6 .... x^2 - x
7 .... x^2 - x - 1
8 .... x^2 + 1
9 .... x^2 + x
10 ... x^2 + x + 1
11 ... x^3
12 ... x^3 - 1
13 ... x^3 - x
The list exemplifies these sequences:
A207187 (multiples of x + 1): 3,5,9,13,...
A207188 (multiples of x): 2,4,6,9,11,13,...
A207189 (multiples of x - 1): 5,6,12,13,...
A207190 (multiples of x^2 + 1): 8,20,25,27,...

Crossrefs

Cf. A206821.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 2000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]]
q2[n_] := p[n - f[k] + 2]
y1 = Table[p[n], {n, 1, 4}];
Do[AppendTo[y1,
  Join[Table[q1[n], {n, f[k], g[k] - 1}],
   Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 10}]
y = Flatten[y1]; (* polynomials over {-1,0,1} *)
TableForm[Table[{n, y[[n]], Factor[y[[n]]]}, {n, 1, 10}]]
Table[y[[n]] /. x -> -1, {n, 1, 300}];
Flatten[Position[%, 0]]  (* A207187 *)
Table[y[[n]] /. x -> 0, {n, 1, 300}] ;
Flatten[Position[%, 0]]  (* A207188 *)
Table[y[[n]] /. x -> 1, {n, 1, 1200}] ;
Flatten[Position[%, 0]]  (* A207189 *)
Table[y[[n]] /. x -> I, {n, 1, 600}] ;
Flatten[Position[%, 0]]  (* A207190 *)

A207189 Numbers matching polynomials y(k,x) that have x-1 as a factor; see Comments.

Original entry on oeis.org

5, 6, 12, 13, 15, 27, 28, 30, 34, 58, 59, 61, 65, 73, 121, 122, 124, 128, 136, 152, 248, 249, 251, 255, 263, 279, 311, 503, 504, 506, 510, 518, 534, 566, 630, 1014, 1015, 1017, 1021, 1029, 1045, 1077, 1141, 1269, 2037, 2038, 2040, 2044, 2052, 2068
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2012

Keywords

Comments

The polynomials y(k,x) range through all monic polynomials with coefficients in {-1,0,1}, ordered as at A206821.

Examples

			The first 13 polynomials:
1 .... 1
2 .... x
3 .... x + 1
4 .... x^2
5 .... x^2 - 1
6 .... x^2 - x
7 .... x^2 - x - 1
8 .... x^2 + 1
9 .... x^2 + x
10 ... x^2 + x + 1
11 ... x^3
12 ... x^3 - 1
13 ... x^3 - x
The list exemplifies these sequences:
A207187 (multiples of x + 1): 3,5,9,13,...
A207188 (multiples of x): 2,4,6,9,11,13,...
A207189 (multiples of x - 1): 5,6,12,13,...
A207190 (multiples of x^2 + 1): 8,20,25,27,...

Crossrefs

Cf. A206821.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 2000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]]
q2[n_] := p[n - f[k] + 2]
y1 = Table[p[n], {n, 1, 4}];
Do[AppendTo[y1,
  Join[Table[q1[n], {n, f[k], g[k] - 1}],
   Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 10}]
y = Flatten[y1]; (* polynomials over {-1,0,1} *)
TableForm[Table[{n, y[[n]], Factor[y[[n]]]}, {n, 1, 10}]]
Table[y[[n]] /. x -> -1, {n, 1, 300}];
Flatten[Position[%, 0]]  (* A207187 *)
Table[y[[n]] /. x -> 0, {n, 1, 300}] ;
Flatten[Position[%, 0]]  (* A207188 *)
Table[y[[n]] /. x -> 1, {n, 1, 1200}] ;
Flatten[Position[%, 0]]  (* A207189 *)
Table[y[[n]] /. x -> I, {n, 1, 600}] ;
Flatten[Position[%, 0]]  (* A207190 *)
Showing 1-4 of 4 results.

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