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 1-6 of 6 results.

A206911 Position of n-th partial sum of the harmonic series when all the partial sums are jointly ranked with the set {log(k+1)}; complement of A206912.

Original entry on oeis.org

2, 5, 8, 11, 13, 16, 19, 22, 24, 27, 30, 33, 36, 38, 41, 44, 47, 49, 52, 55, 58, 61, 63, 66, 69, 72, 74, 77, 80, 83, 86, 88, 91, 94, 97, 100, 102, 105, 108, 111, 113, 116, 119, 122, 125, 127, 130, 133, 136, 138, 141, 142, 143, 144, 145, 146, 147, 148, 149
Offset: 1

Views

Author

Clark Kimberling, Feb 13 2012

Keywords

Comments

Conjecture: the difference sequence of A206911 consists of 2s and 3s, and the ratio (number of 3s)/(number of 2s) tends to a number between 3.5 and 3.6.
Similar conjectures can be stated for difference sequences based on jointly ranked sets, such as A206903, A206906, A206928, A206805, A206812, and A206815.

Examples

			Let S(n)=1+1/2+1/3+...+1/n and L(n)=log(n+1).  Then
L(1)<S(1)<L(2)<L(3)<S(2)<L(4)<L(5)<S(3)<L(6)<..., so that
A206911=(2,5,8,...).

Crossrefs

Cf. A206912, A206815.

Programs

  • Mathematica
    f[n_] := Sum[1/k, {k, 1, n}];  z = 300;
g[n_] := N[Log[n + 1]];
c = Table[f[n], {n, 1, z}];
s = Table[g[n], {n, 1, z}];
j = Sort[Union[c, s]];
p[n_] := Position[j, f[n]]; q[n_] := Position[j, g[n]];
Flatten[Table[p[n], {n, 1, z}]]    (* A206911 *)
Flatten[Table[q[n], {n, 1, z}]]    (* A206912 *)

A206906 n+[ns/r]+[nt/r], where []=floor, r=1/3, s=sqrt(3), t=1/s.

Original entry on oeis.org

7, 15, 23, 30, 38, 47, 55, 62, 70, 78, 87, 94, 102, 110, 117, 126, 134, 142, 149, 157, 166, 174, 181, 189, 197, 206, 213, 221, 229, 236, 245, 253, 261, 268, 276, 285, 293, 300, 308, 316, 325, 332, 340, 348, 355, 364, 372, 380, 387, 395, 404, 412, 419
Offset: 1

Views

Author

Clark Kimberling, Feb 13 2012

Keywords

Comments

The sequences A206906, A206907, A206908 partition the positive integers. To generate them, jointly rank the sets {3n}, {n/sqrt(3)}, {n*sqrt(3)} for n>=1. The positions of 3n in the joint ranking form A206906, and likewise for the other sequences.

Crossrefs

Cf. A206903.

Programs

  • Mathematica
    r = 1/3; s = Sqrt[3]; t = 1/s;
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
Table[a[n], {n, 1, 70}]  (* A206906 *)
Table[b[n], {n, 1, 80}]  (* A206907 *)
Table[c[n], {n, 1, 70}]  (* A206908 *)

A206904 n+[nr/s]+[nt/s], where []=floor, r=3, s=sqrt(3), t=1/s.

Original entry on oeis.org

2, 5, 9, 11, 14, 18, 21, 23, 27, 30, 33, 36, 39, 42, 45, 48, 51, 55, 57, 60, 64, 67, 69, 73, 76, 79, 82, 85, 88, 91, 94, 97, 101, 103, 106, 110, 113, 115, 119, 122, 125, 128, 131, 134, 137, 140, 143, 147, 149, 152, 156, 159, 161, 165, 168, 170, 174, 177
Offset: 1

Views

Author

Clark Kimberling, Feb 13 2012

Keywords

Comments

The sequences A206903, A206904, A206905 partition the positive integers. To generate them, jointly rank the sets {n/3}, {n/sqrt(3)}, {n*sqrt(3)} for n>=1. The positions of n/3 in the joint ranking form A206903, and likewise for the other sequences.

Crossrefs

Cf. A206903, A206905.

Programs

  • Mathematica
    r = 3; s = Sqrt[3]; t = 1/s;
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
Table[a[n], {n, 1, 70}]  (* A206903 *)
Table[b[n], {n, 1, 70}]  (* A206904 *)
Table[c[n], {n, 1, 70}]  (* A206905 *)

A206905 n+[nr/t]+[ns/t], where []=floor, r=3, s=sqrt(3), t=1/s.

Original entry on oeis.org

9, 18, 27, 36, 45, 55, 64, 73, 82, 91, 101, 110, 119, 128, 137, 147, 156, 165, 174, 183, 193, 202, 211, 220, 229, 239, 248, 257, 266, 275, 285, 294, 303, 312, 321, 331, 340, 349, 358, 367, 377, 386, 395, 404, 413, 423, 432, 441, 450, 459, 469, 478
Offset: 1

Views

Author

Clark Kimberling, Feb 13 2012

Keywords

Comments

The sequences A206903, A206904, A206905 partition the positive integers. To generate them, jointly rank the sets {n/3}, {n/sqrt(3)}, {n*sqrt(3)} for n>=1. The positions of n/3 in the joint ranking form A206903, and likewise for the other sequences.

Crossrefs

Cf. A206903, A206904.

Programs

  • Mathematica
    r = 3; s = Sqrt[3]; t = 1/s;
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
Table[a[n], {n, 1, 70}]  (* A206903 *)
Table[b[n], {n, 1, 70}]  (* A206904 *)
Table[c[n], {n, 1, 70}]  (* A206905 *)

A206907 n+[nr/s]+[nt/s], where []=floor, r=1/3, s=sqrt(3), t=1/s.

Original entry on oeis.org

1, 2, 4, 5, 6, 9, 10, 11, 13, 14, 16, 18, 19, 20, 22, 24, 25, 27, 28, 29, 32, 33, 34, 36, 37, 39, 41, 42, 43, 45, 46, 48, 50, 51, 52, 54, 56, 57, 59, 60, 61, 64, 65, 66, 68, 69, 71, 73, 74, 75, 77, 79, 80, 82, 83, 84, 86, 88, 89, 91, 92, 93, 96, 97, 98, 100, 101
Offset: 1

Views

Author

Clark Kimberling, Feb 13 2012

Keywords

Comments

The sequences A206906, A206907, A206908 partition the positive integers. To generate them, jointly rank the sets {3n}, {n/sqrt(3)}, {n*sqrt(3)} for n>=1. The positions of 3n in the joint ranking form A206906, and likewise for the other sequences.

Crossrefs

Cf. A206903.

Programs

  • Mathematica
    r = 1/3; s = Sqrt[3]; t = 1/s;
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
Table[a[n], {n, 1, 70}]  (* A206906 *)
Table[b[n], {n, 1, 80}]  (* A206907 *)
Table[c[n], {n, 1, 70}]  (* A206908 *)

A206908 a(n) = 4*n + floor(n/sqrt(3)).

Original entry on oeis.org

4, 9, 13, 18, 22, 27, 32, 36, 41, 45, 50, 54, 59, 64, 68, 73, 77, 82, 86, 91, 96, 100, 105, 109, 114, 119, 123, 128, 132, 137, 141, 146, 151, 155, 160, 164, 169, 173, 178, 183, 187, 192, 196, 201, 205, 210, 215, 219, 224, 228, 233, 238, 242, 247, 251
Offset: 1

Views

Author

Clark Kimberling, Feb 13 2012

Keywords

Comments

The sequences A206906, A206907, A206908 partition the positive integers. To generate them, jointly rank the sets {3n}, {n/sqrt(3)}, {n*sqrt(3)} for n>=1. The positions of 3n in the joint ranking form A206906, and likewise for the other sequences.
Original name:
n+[nr/t]+[ns/t], where []=floor, r=1/3, s=sqrt(3), t=1/s.

Links

Crossrefs

Cf. A206903.

Programs

  • Maple
    seq(4*n+floor(n/sqrt(3)),n=1..100); # Robert Israel, Oct 18 2020
  • Mathematica
    r = 1/3; s = Sqrt[3]; t = 1/s;
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
Table[a[n], {n, 1, 70}]  (* A206906 *)
Table[b[n], {n, 1, 80}]  (* A206907 *)
Table[c[n], {n, 1, 70}]  (* A206908 *)

Extensions

Name changed by Robert Israel, Oct 18 2020
Showing 1-6 of 6 results.

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