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.

A020904 Positions of 2 in A020903; complement of A191777.

Original entry on oeis.org

2, 4, 7, 9, 11, 13, 16, 18, 22, 24, 27, 29, 31, 34, 37, 39, 42, 44, 46, 48, 51, 53, 56, 58, 61, 63, 65, 67, 69, 72, 74, 76, 79, 81, 84, 86, 88, 90, 92, 94, 97, 99, 101, 103, 106, 108, 111, 113, 115, 117, 121, 123, 126, 128, 130, 132, 135, 137, 139, 142
Offset: 1

Views

Author

Clark Kimberling

Keywords

Examples

			A020903 starts 1,2,1,2,1,1,2,1,2,... in which 2 occupies positions 2, 4, 7, 9, 11,... (Positions refer to offset 1 in A020903).

Links

Crossrefs

Cf. A020903, A191777.

Programs

A191777 Positions of 1 in A020903; complement of A020904.

Original entry on oeis.org

1, 3, 5, 6, 8, 10, 12, 14, 15, 17, 19, 20, 21, 23, 25, 26, 28, 30, 32, 33, 35, 36, 38, 40, 41, 43, 45, 47, 49, 50, 52, 54, 55, 57, 59, 60, 62, 64, 66, 68, 70, 71, 73, 75, 77, 78, 80, 82, 83, 85, 87, 89, 91, 93, 95, 96, 98, 100, 102, 104, 105, 107, 109, 110
Offset: 1

Views

Author

Clark Kimberling, Jun 16 2011

Keywords

Comments

See A020903.

Crossrefs

Cf. A020903, A020904.

Programs

A191770 Lim f(f(...f(n)...)) where f(n) is the fractal sequence A022446.

Original entry on oeis.org

1, 2, 3, 1, 1, 2, 1, 3, 1, 1, 2, 2, 1, 1, 3, 1, 3, 1, 1, 2, 2, 1, 1, 1, 3, 1, 3, 1, 2, 1, 2, 2, 2, 1, 1, 1, 1, 3, 1, 3, 1, 1, 3, 2, 1, 2, 1, 2, 2, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 3, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 3, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3
Offset: 1

Views

Author

Clark Kimberling, Jun 16 2011

Keywords

Comments

Suppose that f(1), f(2), ... is a fractal sequence (such as 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, ..., which contains itself as a proper subsequence - if the first occurrence of each n is deleted, the remaining sequence is identical to the original; see the Wikipedia article for a rigorous definition). Then for each n>=1, the limit L(n) of composites f(f(f...f(n)...)) exists and is one of the numbers in the set {k : f(k)=k}. If f(2)>2, then L(n)=1 for all n; if f(2)=2 and f(3)>3, then L(n) equals 1 or 2 for all n. Examples: A020903, A191770, A191774.

Examples

			Write the counting numbers and A022446 like this:
1..2..3..4..5..6..7..8..9..10..11..12..13..14..15..
1..2..3..1..4..2..5..8..1..4...6...2...7...5...3...
It is then easy to check composites:
1->1, 2->2, 3->3, 4->1, 5->4->1, 6->2, 7->5->4->1,...

Links

Crossrefs

Cf. A020903, A191770, A191771, A191772, A191773, A191774.

Programs

  • Mathematica
    g[n_] :=  Length[Select[Table[FixedPoint[i + PrimePi[#] + 1 &, i + PrimePi[i] + 1], {i, n}], # <= n &]];
f[n_] := PrimePi[NestWhile[g, n, ! PrimeQ[#] && # != 1 &]] + 1;
Array[f, 80]             (* A022446 *)
h[n_] := Nest[f, n, 40]; t = Table[h[n], {n, 1, 300}]  (* A191770 *)
Flatten[Position[t, 1]]  (* A191771 *)
Flatten[Position[t, 2]]  (* A191772 *)
Flatten[Position[t, 3]]  (* A191773 *)

A191774 Lim f(f(...f(n)...)) where f(n) is the Farey fractal sequence, A131967.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1
Offset: 1

Views

Author

Clark Kimberling, Jun 16 2011

Keywords

Comments

Suppose that f(1), f(2), f(3),... is a fractal sequence (a sequence which contains itself as a proper subsequence, such as 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, ...; if the first occurrence of each n is deleted, the remaining sequence is identical to the original; see the Wikipedia article for a rigorous definition). Then for each n>=1, the limit L(n) of composites f(f(f...f(n)...)) exists and is one of the numbers in the set {k : f(k)=k}. Thus, if f(2)>2, then L(n)=1 for all n; if f(2)=2 and f(3)>3, then L(n) is 1 or 2 for all n. Examples: A020903, A191770, A191774

Examples

			Write the counting numbers and A131967 like this:
1..2..3..4..5..6..7..8..9..10..11..12..13..14..15..
1..2..1..3..2..1..4..3..5..2...1...6...4...3...5...
It is then easy to check composites:
1->1, 2->2, 3->1, 4->3->1, 5->2, 6->1, 7->4->3->1,...

Links

Crossrefs

Cf. A020903, A191770, A191775, A191776.

Programs

  • Mathematica
    Farey[n_] := Select[Union@Flatten@Outer[Divide, Range[n + 1] - 1, Range[n]], # <= 1 &];
newpos[n_] := Module[{length = Total@Array[EulerPhi, n] + 1, f1 = Farey[n], f2 = Farey[n - 1], to},
   to = Complement[Range[length], Flatten[Position[f1, #] & /@ f2]];
   ReplacePart[Array[0 &, length],
    Inner[Rule, to, Range[length - Length[to] + 1, length], List]]];
a[n_] := Flatten@Table[Fold[ReplacePart[Array[newpos, i][[#2 + 1]], Inner[Rule, Flatten@Position[Array[newpos, i][[#2 + 1]], 0], #1, List]] &, Array[newpos, i][[1]], Range[i - 1]], {i, n}];
t = a[12]; f[n_] := Part[t, n];
Table[f[n], {n, 1, 100}]          (* A131967 *)
h[n_] := Nest[f, n, 50]
t = Table[h[n], {n, 1, 200}]      (* A191774 *)
s = Flatten[Position[t, 1]]       (* A191775 *)
s = Flatten[Position[t, 2]]       (* A191776 *)
Showing 1-4 of 4 results.

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