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A237270 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(n). 175
1, 3, 2, 2, 7, 3, 3, 12, 4, 4, 15, 5, 3, 5, 9, 9, 6, 6, 28, 7, 7, 12, 12, 8, 8, 8, 31, 9, 9, 39, 10, 10, 42, 11, 5, 5, 11, 18, 18, 12, 12, 60, 13, 5, 13, 21, 21, 14, 6, 6, 14, 56, 15, 15, 72, 16, 16, 63, 17, 7, 7, 17, 27, 27, 18, 12, 18, 91, 19, 19, 30, 30, 20, 8, 8, 20, 90 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

T(n,k) is the number of cells in the k-h region of the n-th set of regions in a diagram of the symmetry of sigma(n), see example.

Row n is a palindromic composition of sigma(n).

Row sums give A000203.

Row n has length A237271(n).

In the row 2n-1 of triangle both the first term and the last term are equal to n.

If n is an odd prime then row n is [m, m], where m = (1 + n)/2.

The connection with A196020 is as follows: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> A239660 --> this sequence.

For the boundary segments in an octant see A237591.

For the boundary segments in a quadrant see A237593.

For the boundary segments in the spiral see also A239660.

For the parts in every quadrant of the spiral see A239931, A239932, A239933, A239934.

We can find the spiral on the terraces of the step pyramid described in A244050. - Omar E. Pol, Dec 07 2016

T(n,k) is also the area of the k-th terrace, from left to right, at the n-th level, starting from the top, of the step pyramid described in A245092 (see Links section). - Omar E. Pol, Aug 14 2018

LINKS

Robert Price, Table of n, a(n) for n = 1..15542 (rows n = 1..5000, flattened)

Hartmut F. W. Hoft, Sample visual documentation for Mathematica code

Omar E. Pol, An infinite step pyramid (A237593, A237270, A262626)

Omar E. Pol, Perspective view of the pyramid (first 16 levels)

Index entries for sequences related to sigma(n)

EXAMPLE

Illustration of the first 19 terms as regions (or parts) of a spiral constructed with the first 11.5 rows of A239660:

.

.                   _ _ _ _ _ _

.                  |  _ _ _ _ _|_ _ _ _ _ 5

.               9 _| |         |_ _ _ _ _|

.             9 _|_ _|                   |_ _ 3

.           _ _| |      _ _ _ _          |_  |

.          |  _ _| 12 _|  _ _ _|_ _ _ 3    |_|_ _ 5

.          | |      _|   |     |_ _ _|         | |

.          | |     |  _ _|           |_ _ 3    | |

.          | |     | |    3 _ _        | |     | |

.          | |     | |     |  _|_ 1    | |     | |

.         _|_|    _|_|    _|_| |_|    _|_|    _|_|    _

.        | |     | |     | |         | |     | |     | |

.        | |     | |     |_|_ _     _| |     | |     | |

.        | |     | |    2  |_ _|_ _|  _|     | |     | |

.        | |     |_|_     2    |_ _ _|7   _ _| |     | |

.        | |    4    |_                 _|  _ _|     | |

.        |_|_ _        |_ _ _ _        |  _|    _ _ _| |

.       6      |_      |_ _ _ _|_ _ _ _| | 15 _|    _ _|

.                |_   4        |_ _ _ _ _|  _|     |

.                  |                       |      _|

.                  |_ _ _ _ _ _            |  _ _|28

.                  |_ _ _ _ _ _|_ _ _ _ _ _| |

.                 6            |_ _ _ _ _ _ _|

.

.

If the sequence does not contain negative terms then its terms can be represented in a quadrant. For the construction of the diagram we use the symmetric Dyck paths of A237593 as shown below:

---------------------------------------------------------------

Triangle         Diagram of the symmetry of sigma (n = 1..24)

---------------------------------------------------------------

.              _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

1;            |_| | | | | | | | | | | | | | | | | | | | | | | |

3;            |_ _|_| | | | | | | | | | | | | | | | | | | | | |

2, 2;         |_ _|  _|_| | | | | | | | | | | | | | | | | | | |

7;            |_ _ _|    _|_| | | | | | | | | | | | | | | | | |

3, 3;         |_ _ _|  _|  _ _|_| | | | | | | | | | | | | | | |

12;           |_ _ _ _|  _| |  _ _|_| | | | | | | | | | | | | |

4, 4;         |_ _ _ _| |_ _|_|    _ _|_| | | | | | | | | | | |

15;           |_ _ _ _ _|  _|     |  _ _ _|_| | | | | | | | | |

5, 3, 5;      |_ _ _ _ _| |      _|_| |  _ _ _|_| | | | | | | |

9, 9;         |_ _ _ _ _ _|  _ _|    _| |    _ _ _|_| | | | | |

6, 6;         |_ _ _ _ _ _| |  _|  _|  _|   |  _ _ _ _|_| | | |

28;           |_ _ _ _ _ _ _| |_ _|  _|  _ _| | |  _ _ _ _|_| |

7, 7;         |_ _ _ _ _ _ _| |  _ _|  _|    _| | |    _ _ _ _|

12, 12;       |_ _ _ _ _ _ _ _| |     |     |  _|_|   |* * * *

8, 8, 8;      |_ _ _ _ _ _ _ _| |  _ _|  _ _|_|       |* * * *

31;           |_ _ _ _ _ _ _ _ _| |  _ _|  _|      _ _|* * * *

9, 9;         |_ _ _ _ _ _ _ _ _| | |_ _ _|      _|* * * * * *

39;           |_ _ _ _ _ _ _ _ _ _| |  _ _|    _|* * * * * * *

10, 10;       |_ _ _ _ _ _ _ _ _ _| | |       |* * * * * * * *

42;           |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|* * * * * * * *

11, 5, 5, 11; |_ _ _ _ _ _ _ _ _ _ _| | |* * * * * * * * * * *

18, 18;       |_ _ _ _ _ _ _ _ _ _ _ _| |* * * * * * * * * * *

12, 12;       |_ _ _ _ _ _ _ _ _ _ _ _| |* * * * * * * * * * *

60;           |_ _ _ _ _ _ _ _ _ _ _ _ _|* * * * * * * * * * *

...

The total number of cells in the first n set of symmetric regions of the diagram equals A024916(n), the sum of all divisors of all positive integers <= n, hence the total number of cells in the n-th set of symmetric regions of the diagram equals sigma(n) = A000203(n).

For n = 9 the 9th row of A237593 is [5, 2, 2, 2, 2, 5] and the 8th row of A237593 is [5, 2, 1, 1, 2, 5] therefore between both symmetric Dyck paths there are three regions (or parts) of sizes [5, 3, 5], so row 9 is [5, 3, 5].

The sum of divisors of 9 is 1 + 3 + 9 = A000203(9) = 13. On the other hand the sum of the parts of the symmetric representation of sigma(9) is 5 + 3 + 5 = 13, equaling the sum of divisors of 9.

For n = 24 the 24th row of A237593 is [13, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 13] and the 23rd row of A237593 is [12, 5, 2, 2, 1, 1, 1, 1, 2, 2, 5, 12] therefore between both symmetric Dyck paths there are only one region (or part) of size 60, so row 24 is 60.

The sum of divisors of 24 is 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = A000203(24) = 60. On the other hand the sum of the parts of the symmetric representation of sigma(24) is 60, equaling the sum of divisors of 24.

Note that the number of *'s in the diagram is 24^2 - A024916(24) = 576 - 491 = A004125(24) = 85.

MATHEMATICA

T[n_, k_] := Ceiling[(n + 1)/k - (k + 1)/2] (* from A235791 *)

path[n_] := Module[{c = Floor[(Sqrt[8n + 1] - 1)/2], h, r, d, rd, k, p = {{0, n}}}, h = Map[T[n, #] - T[n, # + 1] &, Range[c]]; r = Join[h, Reverse[h]]; d = Flatten[Table[{{1, 0}, {0, -1}}, {c}], 1];

rd = Transpose[{r, d}]; For[k = 1, k <= 2c, k++, p = Join[p, Map[Last[p] + rd[[k, 2]] * # &, Range[rd[[k, 1]]]]]]; p]

segments[n_] := SplitBy[Map[Min, Drop[Drop[path[n], 1], -1] - path[n - 1]], # == 0 &]

a237270[n_] := Select[Map[Apply[Plus, #] &, segments[n]], # != 0 &]

Flatten[Map[a237270, Range[40]]] (* data *)

(* Hartmut F. W. Hoft, Jun 23 2014 *)

CROSSREFS

Cf. A000203, A004125, A023196, A024916, A153485, A196020, A231347, A235791, A235796, A236104, A236112, A236540, A237046, A237048, A237271, A237590, A237591, A237593, A239050, A239660, A239663, A239665, A239931, A239932, A239933, A239934, A240020, A240062, A244050, A245092, A262626.

Sequence in context: A089327 A280851 A279391 * A091264 A021760 A092419

Adjacent sequences:  A237267 A237268 A237269 * A237271 A237272 A237273

KEYWORD

nonn,tabf,look

AUTHOR

Omar E. Pol, Feb 19 2014

STATUS

approved

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Last modified October 15 20:04 EDT 2019. Contains 328037 sequences. (Running on oeis4.)