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.

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A092249 Positions of the integers in the standard diagonal enumeration of the rationals (with the integers in the first column and diagonals moving up to the right).

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 18, 22, 28, 32, 42, 46, 58, 64, 72, 80, 96, 102, 120, 128, 140, 150, 172, 180, 200, 212, 230, 242, 270, 278, 308, 324, 344, 360, 384, 396, 432, 450, 474, 490, 530, 542, 584, 604, 628, 650, 696, 712, 754, 774, 806, 830, 882, 900, 940, 964
Offset: 1

Views

Author

Andrew Niedermaier, Feb 20 2004

Keywords

Comments

A002088 without the leading zero. [R. J. Mathar, Jul 20 2009]

Examples

			The first few terms of the full enumeration are 1, 2, 1/2, 3, 1/3, 4, 3/2, 2/3, 1/4, 5, giving a(n) = 1, 2, 4, 6, 10,...
Contribution from _R. J. Mathar_, Jul 20 2009: (Start)
The positions in the first column of the table
....1..1/2..1/3..1/4..1/5..1/6..1/7..1/8..1/9.1/10.1/11.1/12
....2.......2/3.......2/5.......2/7.......2/9......2/11.....
....3..3/2.......3/4..3/5.......3/7..3/8......3/10.3/11.....
....4.......4/3.......4/5.......4/7.......4/9......4/11.....
....5..5/2..5/3..5/4.......5/6..5/7..5/8..5/9......5/11.5/12
....6.................6/5.......6/7................6/11.....
....7..7/2..7/3..7/4..7/5..7/6.......7/8..7/9.7/10.7/11.7/12
....8.......8/3.......8/5.......8/7.......8/9......8/11.....
....9..9/2.......9/4..9/5.......9/7..9/8......9/10.9/11.....
...10......10/3................10/7......10/9.....10/11.....
...11.11/2.11/3.11/4.11/5.11/6.11/7.11/8.11/911/10.....11/12
...12................12/5......12/7...............12/11.....
if scanned along rising antidiagonals, as defined by the ratios A038566(i)/A020653(i). (End)

Links

Crossrefs

Cf. A000010, A002088, A020653, A038566.

Programs

  • Mathematica
    Accumulate[EulerPhi[Range[100]]] (* Paolo Xausa, Oct 19 2023 *)

Extensions

a(11) and a(12) corrected by R. J. Mathar, Jul 20 2009
Incorrect recurrence formula removed by R. J. Mathar, Jul 29 2009
More terms (using A002088) from Michel Marcus, Sep 10 2018

A320040 Consider the Cantor matrix of rational numbers. This sequence reads the numerator, then the denominator as one moves through the matrix along alternate up and down antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 4, 2, 3, 3, 2, 4, 1, 5, 1, 4, 2, 3, 3, 2, 4, 1, 5, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9
Offset: 1

Views

Author

Robert G. Wilson v, Oct 03 2018

Keywords

Comments

This is analogous to reading the rows of a triangle in boustrophedon order.
The antidiagonals are in a certain sense palindromic.

Examples

			The Cantor Matrix begins:
=========================================================================
n\d|   1    2    3    4    5    6    7    8    9    10    11    12    13
---|---------------------------------------------------------------------
1  |  1/1  1/2  1/3  1/4  1/5  1/6  1/7  1/8  1/9  1/10  1/11  1/12  1/13
2  |  2/1  2/2  2/3  2/4  2/5  2/6  2/7  2/8  2/9  2/10  2/11  2/12  2/13
3  |  3/1  3/2  3/3  3/4  3/5  3/6  3/7  3/8  3/9  3/10  3/11  3/12  3/13
4  |  4/1  4/2  4/3  4/4  4/5  4/6  4/7  4/8  4/9  4/10  4/11  4/12  4/13
5  |  5/1  5/2  5/3  5/4  5/5  5/6  5/7  5/8  5/9  5/10  5/11  5/12  5/13
6  |  6/1  6/2  6/3  6/4  6/5  6/6  6/7  6/8  6/9  6/10  6/11  6/12  6/13
7  |  7/1  7/2  7/3  7/4  7/5  7/6  7/7  7/8  7/9  7/10  7/11  7/12  7/13
8  |  8/1  8/2  8/3  8/4  8/5  8/6  8/7  8/8  8/9  8/10  8/11  8/12  8/13
9  |  9/1  9/2  9/3  9/4  9/5  9/6  9/7  9/8  9/9  9/10  9/11  9/12  9/13
10 | 10/1 10/2 10/3 10/4 10/5 10/6 10/7 10/8 10/9 10/10 10/11 10/12 10/13
11 | 11/1 11/2 11/3 11/4 11/5 11/6 11/7 11/8 11/9 11/10 11/11 11/12 11/13
12 | 12/1 12/2 12/3 12/4 12/5 12/6 12/7 12/8 12/9 12/10 12/11 12/12 12/13
13 | 13/1 13/2 13/3 13/4 13/5 13/6 13/7 13/8 13/9 13/10 13/11 13/12 13/13
...

Links

Crossrefs

Cf. A020652, A020653.

Programs

  • Mathematica
    (* to read the Cantor Matrix *) Table[{n, d}, {n, 13}, {d, 13}] // Grid

A373701 Extension of Mahler-Popken complexity to the rationals. The minimal number of 1's required to build the n-th positive rational in the Cantor ordering using only +, /, and *.

Original entry on oeis.org

1, 3, 2, 4, 3, 5, 5, 4, 4, 6, 5, 6, 7, 7, 5, 5, 5, 7, 8, 6, 6, 7, 8, 9, 6, 6, 6, 7, 9, 6, 6, 8, 8, 9, 10, 7, 7, 8, 7, 7, 7, 9, 11, 8, 8, 8, 10, 10, 10, 10, 11, 7, 9, 7, 7, 8, 7, 9, 11, 11, 10, 8, 8, 9, 10, 12, 12, 8, 9, 8, 8, 9, 11, 13, 12, 9, 8, 8, 8, 9, 10
Offset: 1

Views

Author

Adil Soubki, Jun 13 2024

Keywords

Comments

Since we do not require that rationals with denominator 1 be written in the form p/q (i.e., we allow them to be written as p), this reduces to A005245 in the case where q = 1.

Examples

			|    | rational   |  minimal expression         |    a(n) |
|---:|:-----------|:----------------------------|--------:|
|  1 | 1/1        | 1                           |       1 |
|  2 | 1/2        | 1/(1+1)                     |       3 |
|  3 | 2/1        | 1+1                         |       2 |
|  4 | 1/3        | 1/(1+1+1)                   |       4 |
|  5 | 3/1        | 1+1+1                       |       3 |
|  6 | 1/4        | 1/(1+1+1+1)                 |       5 |
|  7 | 2/3        | (1+1)/(1+1+1)               |       5 |
|  8 | 3/2        | 1+(1/(1+1))                 |       4 |
|  9 | 4/1        | 1+1+1+1                     |       4 |
| 10 | 1/5        | 1/(1+1+1+1+1)               |       6 |
| 11 | 5/1        | 1+1+1+1+1                   |       5 |
| 12 | 1/6        | 1/((1+1)*(1+1+1))           |       6 |
| 13 | 2/5        | (1+1)/(1+1+1+1+1)           |       7 |
| 14 | 3/4        | (1+1+1)/(1+1+1+1)           |       7 |
| 15 | 4/3        | 1+(1/(1+1+1))               |       5 |
| 16 | 5/2        | 1+1+(1/(1+1))               |       5 |
| 17 | 6/1        | (1+1)*(1+1+1)               |       5 |
| 18 | 1/7        | 1/((1+1+1)*(1+1) +1)        |       7 |
| 19 | 3/5        | (1+1+1)/(1+1+1+1+1)         |       8 |
| 20 | 5/3        | 1+((1+1)/(1+1+1))           |       6 |
| 21 | 7/1        | (1+1)*(1+1+1)+1             |       6 |
| 22 | 1/8        | 1/((1+1)*(1+1)*(1+1))       |       7 |
| 23 | 2/7        | (1+1)/((1+1)*(1+1+1)+1)     |       8 |
| 24 | 4/5        | (1+1+1+1)/(1+1+1+1+1)       |       9 |
| 25 | 5/4        | 1+(1/(1+1+1+1))             |       6 |
| 26 | 7/2        | 1+1+1+(1/(1+1))             |       6 |
| 27 | 8/1        | (1+1)*(1+1)*(1+1)           |       6 |
| 28 | 1/9        | 1/((1+1+1)*(1+1+1))         |       7 |
| 29 | 3/7        | (1+1+1)/((1+1+1)*(1+1) +1)  |       9 |
| 30 | 7/3        | 1+1+(1/(1+1+1))             |       6 |
| 31 | 9/1        | (1+1+1)*(1+1+1)             |       6 |
| 32 | 1/10       | 1/((1+1+1)*(1+1+1)+1)       |       8 |
| 33 | 2/9        | (1+1)/((1+1+1)*(1+1+1))     |       8 |
| 34 | 3/8        | (1+1+1)/((1+1)*(1+1)*(1+1)) |       9 |
| 35 | 4/7        | (1+1+1+1)/((1+1)*(1+1+1)+1) |      10 |
| 36 | 5/6        | (1/(1+1))+(1/(1+1+1))       |       7 |
| 37 | 6/5        | 1+(1/(1+1+1+1+1))           |       7 |

Links

Crossrefs

Cf. A005245 (Mahler-Popken complexity).
Ordering used: A020652 (Cantor numerators), A020653 (Cantor denominators).
Previous Showing 21-23 of 23 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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