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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A237270 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(n).

Original entry on oeis.org

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
Offset: 1

Views

Author

Omar E. Pol, Feb 19 2014

Keywords

Comments

T(n,k) is the number of cells in the k-th 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 stepped 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 stepped pyramid described in A245092 (see Links section). - Omar E. Pol, Aug 14 2018

Examples

			Illustration of the first 27 terms as regions (or parts) of a spiral constructed with the first 15.5 rows of A239660:
.
.                  _ _ _ _ _ _ _ _
.                 |  _ _ _ _ _ _ _|_ _ _ _ _ _ _ 7
.                 | |             |_ _ _ _ _ _ _|
.             12 _| |                           |
.               |_ _|  _ _ _ _ _ _              |_ _
.         12 _ _|     |  _ _ _ _ _|_ _ _ _ _ 5      |_
.      _ _ _| |    9 _| |         |_ _ _ _ _|         |
.     |  _ _ _|  9 _|_ _|                   |_ _ 3    |_ _ _ 7
.     | |      _ _| |      _ _ _ _          |_  |         | |
.     | |     |  _ _| 12 _|  _ _ _|_ _ _ 3    |_|_ _ 5    | |
.     | |     | |      _|   |     |_ _ _|         | |     | |
.     | |     | |     |  _ _|           |_ _ 3    | |     | |
.     | |     | |     | |    3 _ _        | |     | |     | |
.     | |     | |     | |     |  _|_ 1    | |     | |     | |
.    _|_|    _|_|    _|_|    _|_| |_|    _|_|    _|_|    _|_|    _
.   | |     | |     | |     | |         | |     | |     | |     | |
.   | |     | |     | |     |_|_ _     _| |     | |     | |     | |
.   | |     | |     | |    2  |_ _|_ _|  _|     | |     | |     | |
.   | |     | |     |_|_     2    |_ _ _|7   _ _| |     | |     | |
.   | |     | |    4    |_                 _|  _ _|     | |     | |
.   | |     |_|_ _        |_ _ _ _        |  _|    _ _ _| |     | |
.   | |    6      |_      |_ _ _ _|_ _ _ _| | 15 _|    _ _|     | |
.   |_|_ _ _        |_   4        |_ _ _ _ _|  _|     |    _ _ _| |
.  8      | |_ _      |                       |      _|   |  _ _ _|
.         |_    |     |_ _ _ _ _ _            |  _ _|28  _| |
.           |_  |_    |_ _ _ _ _ _|_ _ _ _ _ _| |      _|  _|
.          8  |_ _|  6            |_ _ _ _ _ _ _|  _ _|  _|
.                 |                               |  _ _|  31
.                 |_ _ _ _ _ _ _ _                | |
.                 |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _| |
.                8                |_ _ _ _ _ _ _ _ _|
.
.
[For two other drawings of the spiral see the links. - _N. J. A. Sloane_, Nov 16 2020]
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.
From _Omar E. Pol_, Nov 22 2020: (Start)
Also consider the infinite double-staircases diagram defined in A335616 (see the theorem).
For n = 15 the diagram with first 15 levels looks like this:
.
Level                         "Double-staircases" diagram
.                                          _
1                                        _|1|_
2                                      _|1 _ 1|_
3                                    _|1  |1|  1|_
4                                  _|1   _| |_   1|_
5                                _|1    |1 _ 1|    1|_
6                              _|1     _| |1| |_     1|_
7                            _|1      |1  | |  1|      1|_
8                          _|1       _|  _| |_  |_       1|_
9                        _|1        |1  |1 _ 1|  1|        1|_
10                     _|1         _|   | |1| |   |_         1|_
11                   _|1          |1   _| | | |_   1|          1|_
12                 _|1           _|   |1  | |  1|   |_           1|_
13               _|1            |1    |  _| |_  |    1|            1|_
14             _|1             _|    _| |1 _ 1| |_    |_             1|_
15            |1              |1    |1  | |1| |  1|    1|              1|
.
Starting from A196020 and after the algorithm described in A280850 and A296508 applied to the above diagram we have a new diagram as shown below:
.
Level                             "Ziggurat" diagram
.                                          _
6                                         |1|
7                            _            | |            _
8                          _|1|          _| |_          |1|_
9                        _|1  |         |1   1|         |  1|_
10                     _|1    |         |     |         |    1|_
11                   _|1      |        _|     |_        |      1|_
12                 _|1        |       |1       1|       |        1|_
13               _|1          |       |         |       |          1|_
14             _|1            |      _|    _    |_      |            1|_
15            |1              |     |1    |1|    1|     |              1|
.
The 15th row
of A249351 :  [1,1,1,1,1,1,1,1,0,0,0,1,1,1,2,1,1,1,0,0,0,1,1,1,1,1,1,1,1]
The 15th row
of triangle:  [              8,            8,            8              ]
The 15th row
of A296508:   [              8,      7,    1,    0,      8              ]
The 15th row
of A280851    [              8,      7,    1,            8              ]
.
More generally, for n >= 1, it appears there is the same correspondence between the original diagram of the symmetric representation of sigma(n) and the "Ziggurat" diagram of n.
For the definition of subparts see A239387 and also A296508, A280851. (End)

Links

Crossrefs

Cf. A000203, A004125, A023196, A024916, A153485, A196020, A221529, A231347, A235791, A235796, A236104, A236112, A236540, A237046, A237048, A237271, A237590, A237591, A237593, A239050, A239660, A239663, A239665, A239931, A239932, A239933, A239934, A240020, A240062, A244050, A245092, A249351, A262626, A280850, A280851, A296508, A335616, A340035, A384149.

Programs

  • 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 *)

Formula

T(n, k) = (A384149(n, k) + A384149(n, m+1-k))/2, where m = A237271(n) is the row length. (conjectured) - Peter Munn, Jun 01 2025

Extensions

Drawing of the spiral extended by Omar E. Pol, Nov 22 2020

A244050 Partial sums of A243980.

Original entry on oeis.org

4, 20, 52, 112, 196, 328, 492, 716, 992, 1340, 1736, 2244, 2808, 3468, 4224, 5104, 6056, 7164, 8352, 9708, 11192, 12820, 14544, 16508, 18596, 20852, 23268, 25908, 28668, 31716, 34892, 38320, 41940, 45776, 49804, 54196, 58740, 63524, 68532, 73900
Offset: 1

Views

Author

Omar E. Pol, Jun 18 2014

Keywords

Comments

a(n) is also the volume of a special stepped pyramid with n levels related to the symmetric representation of sigma. Note that starting at the top of the pyramid, the total area of the horizontal regions at the n-th level is equal to A239050(n), and the total area of the vertical regions at the n-th level is equal to 8*n.
From Omar E. Pol, Sep 19 2015: (Start)
Also, consider that the area of the central square in the top of the pyramid is equal to 1, so the total area of the horizontal regions at the n-th level starting from the top is equal to sigma(n) = A000203(n), and the total area of the vertical regions at the n-th level is equal to 2*n.
Also note that this stepped pyramid can be constructed with four copies of the stepped pyramid described in A245092 back-to-back (one copy in every quadrant). (End)
From Omar E. Pol, Jan 20 2021: (Start)
Convolution of A000203 and the nonzero terms of A008586.
Convolution of A074400 and the nonzero terms of A005843.
Convolution of A340793 and the nonzero terms of A046092.
Convolution of A239050 and A000027.
(End)

Examples

			From _Omar E. Pol_, Aug 29 2015: (Start)
Illustration of the top view of the stepped pyramid with 16 levels. The pyramid is formed of 5104 unit cubes:
.                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.             _ _| |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  | |_ _
.           _|  _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _  |_
.         _|  _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_  |_
.        |  _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_  |
.   _ _ _| |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  | |_ _ _
.  |  _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _  |
.  | | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | | |
.  | | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | | |
.  | | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | | |
.  | | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | | |
.  | | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | | |
.  | | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | | |
.  | | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | | |
.  | | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | | |
.  | | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | | |
.  | | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | | |
.  | | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | | |
.  | |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_| |
.  |_ _ _  | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |  _ _ _|
.        | |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _| |
.        |_  |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|  _|
.          |_  |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|  _|
.            |_ _  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |  _ _|
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Note that the above diagram contains a hidden pattern, simpler, which emerges from the front view of every corner of the stepped pyramid.
For more information about the hidden pattern see A237593 and A245092.
(End)

Links

Crossrefs

Cf. A000203, A024916, A046092, A008586, A175254, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A239050, A239660, A239931, A239932, A239933, A239934, A243980, A245092, A262626, A340793.

Programs

  • Magma
    [4*(&+[(n-k+1)*DivisorSigma(1,k): k in [1..n]]): n in [1..40]]; // G. C. Greubel, Apr 07 2019
  • Mathematica
    a[n_] := 4 Sum[(n - k + 1) DivisorSigma[1, k], {k, n}]; Array[a, 40] (* Robert G. Wilson v, Aug 06 2018 *)
Nest[Accumulate,4*DivisorSigma[1,Range[50]],2] (* Harvey P. Dale, Sep 07 2022 *)
  • PARI
    a(n) = 4*sum(k=1, n, sigma(k)*(n-k+1)); \\ Michel Marcus, Aug 07 2018
  • Python
    from math import isqrt
def A244050(n): return (((s:=isqrt(n))**2*(s+1)*((s+1)*((s<<1)+1)-6*(n+1))>>1) + sum((q:=n//k)*(-k*(q+1)*(3*k+(q<<1)+1)+3*(n+1)*((k<<1)+q+1)) for k in range(1,s+1))<<1)//3 # Chai Wah Wu, Oct 22 2023
  • Sage
    [4*sum(sigma(k)*(n-k+1) for k in (1..n)) for n in (1..40)] # G. C. Greubel, Apr 07 2019

Formula

a(n) = 4*A175254(n).

A239660 Triangle read by rows in which row n lists two copies of the n-th row of triangle A237593.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 3, 1, 1, 3, 3, 1, 1, 3, 3, 2, 2, 3, 3, 2, 2, 3, 4, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 4, 4, 2, 1, 1, 2, 4, 4, 2, 1, 1, 2, 4, 5, 2, 1, 1, 2, 5, 5, 2, 1, 1, 2, 5, 5, 2, 2, 2, 2, 5, 5, 2, 2, 2, 2, 5, 6, 2, 1, 1, 1, 1, 2, 6, 6, 2, 1, 1, 1, 1, 2, 6, 6, 3, 1, 1, 1, 1, 3, 6, 6, 3, 1, 1, 1, 1, 3, 6
Offset: 1

Views

Author

Omar E. Pol, Mar 24 2014

Keywords

Comments

For the construction of this sequence also we can start from A235791.
This sequence can be interpreted as an infinite Dyck path: UDUDUUDD...
Also we use this sequence for the construction of a spiral in which the arms in the quadrants give the symmetric representation of sigma, see example.
We can find the spiral (mentioned above) on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016
The spiral has the property that the sum of the parts in the quadrants 1 and 3, divided by the sum of the parts in the quadrants 2 and 4, converges to 3/5. - Omar E. Pol, Jun 10 2019

Examples

			Triangle begins (first 15.5 rows):
1, 1, 1, 1;
2, 2, 2, 2;
2, 1, 1, 2, 2, 1, 1, 2;
3, 1, 1, 3, 3, 1, 1, 3;
3, 2, 2, 3, 3, 2, 2, 3;
4, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 4;
4, 2, 1, 1, 2, 4, 4, 2, 1, 1, 2, 4;
5, 2, 1, 1, 2, 5, 5, 2, 1, 1, 2, 5;
5, 2, 2, 2, 2, 5, 5, 2, 2, 2, 2, 5;
6, 2, 1, 1, 1, 1, 2, 6, 6, 2, 1, 1, 1, 1, 2, 6;
6, 3, 1, 1, 1, 1, 3, 6, 6, 3, 1, 1, 1, 1, 3, 6;
7, 2, 2, 1, 1, 2, 2, 7, 7, 2, 2, 1, 1, 2, 2, 7;
7, 3, 2, 1, 1, 2, 3, 7, 7, 3, 2, 1, 1, 2, 3, 7;
8, 3, 1, 2, 2, 1, 3, 8, 8, 3, 1, 2, 2, 1, 3, 8;
8, 3, 2, 1, 1, 1, 1, 2, 3, 8, 8, 3, 2, 1, 1, 1, 1, 2, 3, 8;
9, 3, 2, 1, 1, 1, 1, 2, 3, 9, ...
.
Illustration of initial terms as an infinite Dyck path (row n = 1..4):
.
.                            /\/\    /\/\
.       /\  /\  /\/\  /\/\  /    \  /    \
.  /\/\/  \/  \/    \/    \/      \/      \
.
.
Illustration of initial terms for the construction of a spiral related to sigma:
.
.  row 1     row 2          row 3           row 4
.                                          _ _ _
.                                               |_
.             _ _                                 |
.   _ _      |                                    |
.  |   |     |                                    |
.            |         |           |              |
.            |_ _      |_         _|              |
.                        |_ _ _ _|               _|
.                                          _ _ _|
.
.[1,1,1,1] [2,2,2,2] [2,1,1,2,2,1,1,2] [3,1,1,3,3,1,1,3]
.
The first 2*A003056(n) terms of the n-th row are represented in the A010883(n-1) quadrant and the last 2*A003056(n) terms of the n-th row are represented in the A010883(n) quadrant.
.
Illustration of the spiral constructed with the first 15.5 rows of triangle:
.
.               12 _ _ _ _ _ _ _ _
.                 |  _ _ _ _ _ _ _|_ _ _ _ _ _ _ 7
.                 | |             |_ _ _ _ _ _ _|
.                _| |                           |
.               |_ _|9 _ _ _ _ _ _              |_ _
.         12 _ _|     |  _ _ _ _ _|_ _ _ _ _ 5      |_
.      _ _ _| |      _| |         |_ _ _ _ _|         |
.     |  _ _ _|  9 _|_ _|                   |_ _ 3    |_ _ _ 7
.     | |      _ _| |   12 _ _ _ _          |_  |         | |
.     | |     |  _ _|    _|  _ _ _|_ _ _ 3    |_|_ _ 5    | |
.     | |     | |      _|   |     |_ _ _|         | |     | |
.     | |     | |     |  _ _|           |_ _ 3    | |     | |
.     | |     | |     | |    3 _ _        | |     | |     | |
.     | |     | |     | |     |  _|_ 1    | |     | |     | |
.    _|_|    _|_|    _|_|    _|_| |_|    _|_|    _|_|    _|_|    _
.   | |     | |     | |     | |         | |     | |     | |     | |
.   | |     | |     | |     |_|_ _     _| |     | |     | |     | |
.   | |     | |     | |    2  |_ _|_ _|  _|     | |     | |     | |
.   | |     | |     |_|_     2    |_ _ _|    _ _| |     | |     | |
.   | |     | |    4    |_               7 _|  _ _|     | |     | |
.   | |     |_|_ _        |_ _ _ _        |  _|    _ _ _| |     | |
.   | |    6      |_      |_ _ _ _|_ _ _ _| |    _|    _ _|     | |
.   |_|_ _ _        |_   4        |_ _ _ _ _|  _|     |    _ _ _| |
.  8      | |_ _      |                     15|      _|   |  _ _ _|
.         |_    |     |_ _ _ _ _ _            |  _ _|    _| |
.        8  |_  |_    |_ _ _ _ _ _|_ _ _ _ _ _| |      _|  _|
.             |_ _|  6            |_ _ _ _ _ _ _|  _ _|  _|
.                 |                             28|  _ _|
.                 |_ _ _ _ _ _ _ _                | |
.                 |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _| |
.                8                |_ _ _ _ _ _ _ _ _|
.                                                    31
.
The diagram contains A237590(16) = 27 parts.
The total area (also the total number of cells) in the n-th arm of the spiral is equal to sigma(n) = A000203(n), considering every quadrant and the axes x and y. (checked by hand up to row n = 128). The parts of the spiral are in A237270: 1, 3, 2, 2, 7...
Diagram extended by _Omar E. Pol_, Aug 23 2018

Links

Crossrefs

Row n has length 4*A003056(n).
The sum of row n is equal to 4*n = A008586(n).
Row n is a palindromic composition of 4*n = A008586(n).
Both column 1 and right border are A008619, n >= 1.
The connection between A196020 and A237270 is as follows: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> this sequence --> A237270.
Cf. A000203, A000217, A003056, A008619, A010883, A112610, A193553, A237048, A237271, A237590, A239052, A239053, A239663, A239665, A239931, A239932, A239933, A239934, A240020, A240062, A244050, A245092, A262626, A296508, A299778.

A239931 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(4n-3).

Original entry on oeis.org

1, 3, 3, 5, 3, 5, 7, 7, 9, 9, 11, 5, 5, 11, 13, 5, 13, 15, 15, 17, 7, 7, 17, 19, 19, 21, 21, 23, 32, 23, 25, 7, 25, 27, 27, 29, 11, 11, 29, 31, 31, 33, 9, 9, 33, 35, 13, 13, 35, 37, 37, 39, 18, 39, 41, 15, 9, 15, 41, 43, 11, 11, 43, 45, 45, 47, 17, 17, 47, 49, 49, 51, 51, 53, 43, 43, 53, 55, 55, 57, 57, 59, 21, 22, 21, 59, 61, 11, 61, 63, 15, 15, 63
Offset: 1

Views

Author

Omar E. Pol, Mar 29 2014

Keywords

Comments

Row n is a palindromic composition of sigma(4n-3).
Row n is also the row 4n-3 of A237270.
Row n has length A237271(4n-3).
Row sums give A112610.
Also row n lists the parts of the symmetric representation of sigma in the n-th arm of the first quadrant of the spiral described in A239660, see example.
For the parts of the symmetric representation of sigma(4n-2), see A239932.
For the parts of the symmetric representation of sigma(4n-1), see A239933.
For the parts of the symmetric representation of sigma(4n), see A239934.
We can find the spiral (mentioned above) on the terraces of the pyramid described in A244050. - Omar E. Pol, Dec 06 2016

Examples

			The irregular triangle begins:
   1;
   3,  3;
   5,  3,  5;
   7,  7;
   9,  9;
  11,  5,  5, 11;
  13,  5, 13;
  15, 15;
  17,  7,  7, 17;
  19, 19;
  21, 21;
  23, 32, 23;
  25,  7, 25;
  27, 27;
  29, 11, 11, 29;
  31, 31;
  ...
Illustration of initial terms in the first quadrant of the spiral described in A239660:
.
.     _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 15
.    |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.                                  |
.                                  |
.     _ _ _ _ _ _ _ _ _ _ _ _ _ 13 |
.    |_ _ _ _ _ _ _ _ _ _ _ _ _|   |
.                              |   |_ _ _
.                              |         |
.     _ _ _ _ _ _ _ _ _ _ _ 11 |         |_
.    |_ _ _ _ _ _ _ _ _ _ _|   |_ _ _      |_
.                          |         |_ _ 5  |_
.                          |         |_  |_    |_ _
.     _ _ _ _ _ _ _ _ _ 9  |_ _ _      |_  |       |
.    |_ _ _ _ _ _ _ _ _|   |_ _  |_ 5    |_|_      |
.                      |       |_ _|_ 5      |     |_ _ _ _ _ _ 15
.                      |           | |_      |               | |
.     _ _ _ _ _ _ _ 7  |_ _        |_  |     |_ _ _ _ _ 13   | |
.    |_ _ _ _ _ _ _|       |_        | |             | |     | |
.                  |         |_      |_|_ _ _ _ 11   | |     | |
.                  |_ _        |             | |     | |     | |
.     _ _ _ _ _ 5      |_      |_ _ _ _ 9    | |     | |     | |
.    |_ _ _ _ _|         |           | |     | |     | |     | |
.              |_ _ 3    |_ _ _ 7    | |     | |     | |     | |
.              |_  |         | |     | |     | |     | |     | |
.     _ _ _ 3    |_|_ _ 5    | |     | |     | |     | |     | |
.    |_ _ _|         | |     | |     | |     | |     | |     | |
.          |_ _ 3    | |     | |     | |     | |     | |     | |
.            | |     | |     | |     | |     | |     | |     | |
.     _ 1    | |     | |     | |     | |     | |     | |     | |
.    |_|     |_|     |_|     |_|     |_|     |_|     |_|     |_|
.
For n = 7 we have that 4*7-3 = 25 and the 25th row of A237593 is [13, 5, 3, 1, 2, 1, 1, 2, 1, 3, 5, 13] and the 24th row of A237593 is [13, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 13] therefore between both Dyck paths there are three regions (or parts) of sizes [13, 5, 13], so row 7 is [13, 5, 13].
The sum of divisors of 25 is 1 + 5 + 25 = A000203(25) = 31. On the other hand the sum of the parts of the symmetric representation of sigma(25) is 13 + 5 + 13 = 31, equaling the sum of divisors of 25.

Crossrefs

Cf. A000203, A112610, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A239660, A239932-A239934, A244050, A245092, A262626.

A239934 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(4n).

Original entry on oeis.org

7, 15, 28, 31, 42, 60, 56, 63, 91, 90, 42, 42, 124, 49, 49, 120, 168, 127, 63, 63, 195, 70, 70, 186, 224, 180, 84, 84, 252, 217, 210, 280, 248, 105, 105, 360, 112, 112, 255
Offset: 1

Views

Author

Omar E. Pol, Mar 29 2014

Keywords

Comments

Row n is a palindromic composition of sigma(4n).
Row n is also the row 4n of A237270.
Row n has length A237271(4n).
Row sums give A193553.
First differs from A193553 at a(11).
Also row n lists the parts of the symmetric representation of sigma in the n-th arm of the fourth quadrant of the spiral described in A239660, see example.
For the parts of the symmetric representation of sigma(4n-3), see A239931.
For the parts of the symmetric representation of sigma(4n-2), see A239932.
For the parts of the symmetric representation of sigma(4n-1), see A239933.
We can find the spiral (mentioned above) on the terraces of the pyramid described in A244050. - Omar E. Pol, Dec 06 2016

Examples

			The irregular triangle begins:
    7;
   15;
   28;
   31;
   42;
   60;
   56;
   63;
   91;
   90;
   42, 42;
  124;
   49, 49;
  120;
  168;
  ...
Illustration of initial terms in the fourth quadrant of the spiral described in A239660:
.
.           7       15      28      31      42      60      56      63
.           _       _       _       _       _       _       _       _
.          | |     | |     | |     | |     | |     | |     | |     | |
.         _| |     | |     | |     | |     | |     | |     | |     | |
.     _ _|  _|     | |     | |     | |     | |     | |     | |     | |
.    |_ _ _|    _ _| |     | |     | |     | |     | |     | |     | |
.             _|  _ _|     | |     | |     | |     | |     | |     | |
.            |  _|    _ _ _| |     | |     | |     | |     | |     | |
.     _ _ _ _| |    _|    _ _|     | |     | |     | |     | |     | |
.    |_ _ _ _ _|  _|     |    _ _ _| |     | |     | |     | |     | |
.                |      _|   |  _ _ _|     | |     | |     | |     | |
.                |  _ _|    _| |    _ _ _ _| |     | |     | |     | |
.     _ _ _ _ _ _| |      _|  _|   |  _ _ _ _|     | |     | |     | |
.    |_ _ _ _ _ _ _|  _ _|  _|  _ _| |    _ _ _ _ _| |     | |     | |
.                    |  _ _|  _|    _|   |    _ _ _ _|     | |     | |
.                    | |     |     |  _ _|   |    _ _ _ _ _| |     | |
.     _ _ _ _ _ _ _ _| |  _ _|  _ _|_|       |   |  _ _ _ _ _|     | |
.    |_ _ _ _ _ _ _ _ _| |  _ _|  _|      _ _|   | |    _ _ _ _ _ _| |
.                        | |     |      _|    _ _| |   |  _ _ _ _ _ _|
.                        | |  _ _|    _|  _ _|  _ _|   | |
.     _ _ _ _ _ _ _ _ _ _| | |       |   |    _|    _ _| |
.    |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|  _|  _|     |  _ _|
.                            | |       |  _|      _| |
.                            | |  _ _ _| |      _|  _|
.     _ _ _ _ _ _ _ _ _ _ _ _| | |  _ _ _|  _ _|  _|
.    |_ _ _ _ _ _ _ _ _ _ _ _ _| | |       |  _ _|
.                                | |  _ _ _| |
.                                | | |  _ _ _|
.     _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | |
.    |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.                                    | |
.                                    | |
.     _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.    |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
For n = 7 we have that 4*7 = 28 and the 28th row of A237593 is [15, 5, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 5, 15] and the 27th row of A237593 is [14, 5, 3, 2, 1, 2, 2, 1, 2, 3, 5, 14] therefore between both Dyck paths there are only one region (or part) of size 56, so row 7 is 56.
The sum of divisors of 28 is 1 + 2 + 4 + 7 + 14 + 28 = A000203(28) = 56. On the other hand the sum of the parts of the symmetric representation of sigma(28) is 56, equaling the sum of divisors of 28.
For n = 11 we have that 4*11 = 44 and the 44th row of A237593 is [23, 8, 4, 3, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 8, 23] and the 43rd row of A237593 is [22, 8, 4, 3, 2, 1, 2, 1, 1, 2, 1, 2, 3, 4, 8, 23] therefore between both Dyck paths there are two regions (or parts) of sizes [42, 42], so row 11 is [42, 42].
The sum of divisors of 44 is 1 + 2 + 4 + 11 + 22 + 44 = A000203(44) = 84. On the other hand the sum of the parts of the symmetric representation of sigma(44) is 42 + 42 = 84, equaling the sum of divisors of 44.

Crossrefs

Cf. A000203, A193553, A196020, A236104, A235791, A237048, A237270, A237271, A237591, A237593, A239660, A239931, A239932, A239933, A244050, A245092, A262626.

A239932 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(4n-2).

Original entry on oeis.org

3, 12, 9, 9, 12, 12, 39, 18, 18, 21, 21, 72, 27, 27, 30, 30, 96, 36, 36, 39, 15, 39, 120, 45, 45, 48, 48, 144, 54, 36, 54, 57, 57, 84, 84, 63, 63, 66, 66, 234, 72, 72, 75, 21, 75, 108, 108, 81, 81, 84, 48, 84, 120, 120, 90, 90, 93, 93, 312
Offset: 1

Views

Author

Omar E. Pol, Mar 29 2014

Keywords

Comments

Row n is a palindromic composition of sigma(4n-2).
Row n is also the row 4n-2 of A237270.
Row n has length A237271(4n-2).
Row sums give A239052.
Also row n lists the parts of the symmetric representation of sigma in the n-th arm of the second quadrant of the spiral described in A239660, see example.
For the parts of the symmetric representation of sigma(4n-3), see A239931.
For the parts of the symmetric representation of sigma(4n-1), see A239933.
For the parts of the symmetric representation of sigma(4n), see A239934.
We can find the spiral (mentioned above) on the terraces of the pyramid described in A244050. - Omar E. Pol, Dec 06 2016

Examples

			The irregular triangle begins:
3;
12;
9, 9;
12, 12;
39;
18, 18;
21, 21;
72;
27, 27;
30, 30;
96;
36, 36;
39, 15, 39;
120;
45, 45;
48, 48;
...
Illustration of initial terms in the second quadrant of the spiral described in A239660:
.                                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.                                | |
.                                | |
.                                | |  _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                           _ _ _| | |  _ _ _ _ _ _ _ _ _ _ _ _ _|
.                          |       | | |
.                       _ _|  _ _ _| | |
.                  72 _|     |       | |  _ _ _ _ _ _ _ _ _ _ _ _
.                   _|      _| 21 _ _| | |  _ _ _ _ _ _ _ _ _ _ _|
.                  |      _|     |_ _ _| | |
.               _ _|    _|    _ _|       | |
.              |    _ _|    _|     18 _ _| |  _ _ _ _ _ _ _ _ _ _
.              |   |       |         |_ _ _| |  _ _ _ _ _ _ _ _ _|
.     _ _ _ _ _|   | 21 _ _|        _|       | |
.    |  _ _ _ _ _ _|   | |        _|      _ _| |
.    | |      _ _ _ _ _| | 18 _ _|       |     |  _ _ _ _ _ _ _ _
.    | |     |  _ _ _ _ _|   | |     39 _|  _ _| |  _ _ _ _ _ _ _|
.    | |     | |      _ _ _ _| |    _ _|  _|     | |
.    | |     | |     |  _ _ _ _|   |    _|   12 _| |
.    | |     | |     | |      _ _ _|   |       |_ _|  _ _ _ _ _ _
.    | |     | |     | |     |  _ _ _ _| 12 _ _|     |  _ _ _ _ _|
.    | |     | |     | |     | |      _ _ _| |    9 _| |
.    | |     | |     | |     | |     |  _ _ _|  9 _|_ _|
.    | |     | |     | |     | |     | |      _ _| |      _ _ _ _
.    | |     | |     | |     | |     | |     |  _ _| 12 _|  _ _ _|
.    | |     | |     | |     | |     | |     | |      _|   |
.    | |     | |     | |     | |     | |     | |     |  _ _|
.    | |     | |     | |     | |     | |     | |     | |    3 _ _
.    | |     | |     | |     | |     | |     | |     | |     |  _|
.    |_|     |_|     |_|     |_|     |_|     |_|     |_|     |_|
.
For n = 7 we have that 4*7-2 = 26 and the 26th row of A237593 is [14, 5, 2, 2, 2, 1, 1, 2, 2, 2, 5, 14] and the 25th row of A237593 is [13, 5, 3, 1, 2, 1, 1, 2, 1, 3, 5, 13] therefore between both Dyck paths there are two regions (or parts) of sizes [21, 21], so row 7 is [21, 21].
The sum of divisors of 26 is 1 + 2 + 13 + 26 = A000203(26) = 42. On the other hand the sum of the parts of the symmetric representation of sigma(26) is 21 + 21 = 42, equaling the sum of divisors of 26.

Crossrefs

Cf. A000203, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A239052, A239660, A239931, A239933, A239934, A244050, A245092, A262626.

A319529 Odd numbers that have middle divisors.

Original entry on oeis.org

1, 9, 15, 25, 35, 45, 49, 63, 77, 81, 91, 99, 117, 121, 135, 143, 153, 165, 169, 187, 195, 209, 221, 225, 231, 247, 255, 273, 285, 289, 299, 315, 323, 325, 345, 357, 361, 375, 391, 399, 405, 425, 435, 437, 441, 459, 475, 483, 493, 513, 525, 527, 529, 551, 561, 567, 575, 589, 609, 621, 625, 627, 651
Offset: 1

Views

Author

Omar E. Pol, Sep 23 2018

Keywords

Comments

Odd numbers k such that the symmetric representation of sigma(k) has an odd number of parts.
From Felix Fröhlich, Sep 25 2018: (Start)
For the definition of middle divisors, see A067742.
Let t be a term of A005408. Then t is in this sequence iff A067742(t) != 0. (End)
From Hartmut F. W. Hoft, May 24 2022: (Start)
By Theorem 1 (iii) in A067742, the number of middle divisors of a(n) equals the width of the symmetric representation of sigma(a(n)) on the diagonal which equals the triangle entry A249223(n, A003056(n)).
All terms in sequence A016754 have an odd number of middle divisors, forming a subsequence of this sequence; A016754(18) = a(116) = 1225 = 5^2 * 7^2 is the smallest number in A016754 with 3 middle divisors: 25, 35, 49.
Sequence A259417 is a subsequence of this sequence and of A320137 since an even power of a prime has a single middle divisor.
The maximum widths of the center part of the symmetric representation of sigma(a(n)), SRS(a(n)), need not occur at the diagonal. For example, a(304) = 3^3 * 5^3 = 3375, SRS(3375) has 3 parts, its center part has maximum width 3 while its width at the diagonal equals 2 = A067742(3375), and divisors 45 and 75 are the two middle divisors of a(304). (End)

Examples

			9 is in the sequence because it's an odd number and the symmetric representation of sigma(9) = 13 has an odd number of parts (more exactly three parts), as shown below:
.
.     _ _ _ _ _ 5
.    |_ _ _ _ _|
.              |_ _ 3
.              |_  |
.                |_|_ _ 5
.                    | |
.                    | |
.                    | |
.                    | |
.                    |_|
.

Crossrefs

Intersection of A005408 and A071562.
For more information about the diagram see A237593.
Cf. A000203, A067742, A071090, A236104, A237270, A237271, A239931, A239933, A240542, A245092, A280919, A281007, A296508, A299761, A299777, A303297, A319796, A319801, A319802.
Cf. A016754, A237048, A249223, A259417, A320137.

Programs

  • Mathematica
    middleDiv[n_] := Select[Divisors[n], Sqrt[n/2]<=#Hartmut F. W. Hoft, May 24 2022 *)
  • Python
    from itertools import islice, count
from sympy import divisors
def A319529_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(1,startvalue+1-(startvalue&1)),2):
        if any((k <= 2*d**2 < 4*k for d in divisors(k,generator=True))):
            yield k
A319529_list = list(islice(A319529_gen(startvalue=11),40)) # Chai Wah Wu, Jun 09 2022

A319801 Odd numbers without middle divisors.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 79, 83, 85, 87, 89, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 123, 125, 127, 129, 131, 133, 137, 139, 141, 145, 147, 149, 151, 155, 157, 159, 161, 163, 167, 171, 173, 175, 177
Offset: 1

Views

Author

Omar E. Pol, Sep 28 2018

Keywords

Comments

Odd numbers k such that the symmetric representation of sigma(k) has an even number of parts.
All odd primes (A065091) are in the sequence.
For the definition of middle divisors, see A067742.
For more information about the symmetric representation of sigma(k) see A237593.

Examples

			21 is in the sequence because it's an odd number and the symmetric representation of sigma(21) = 32 has an even number of parts (more exactly four parts), as shown below:
.     _ _ _ _ _ _ _ _ _ _ _ 11
.    |_ _ _ _ _ _ _ _ _ _ _|
.                          |
.                          |
.                          |_ _ _ 5
.                          |_ _  |_
.                              |_ _|_
.                                  | |_ 5
.                                  |_  |
.                                    | |
.                                    |_|_ _ _ _ 11
.                                            | |
.                                            | |
.                                            | |
.                                            | |
.                                            | |
.                                            | |
.                                            | |
.                                            | |
.                                            | |
.                                            | |
.                                            |_|
.

Crossrefs

Intersection of A005408 and A071561.
Cf. A000203, A065091, A067742, A071090, A071562, A236104, A237270, A237271, A237593, A239931, A239933, A240542, A245092, A280919, A281007, A296508, A299761, A299777, A303297, A319529, A319796, A319802.

A317303 Numbers k such that both Dyck paths of the symmetric representation of sigma(k) have a central peak.

Original entry on oeis.org

2, 7, 8, 9, 16, 17, 18, 19, 20, 29, 30, 31, 32, 33, 34, 35, 46, 47, 48, 49, 50, 51, 52, 53, 54, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 154, 155, 156, 157, 158, 159, 160
Offset: 1

Views

Author

Omar E. Pol, Aug 27 2018

Keywords

Comments

Also triangle read by rows which gives the odd-indexed rows of triangle A014132.
There are no triangular number (A000217) in this sequence.
For more information about the symmetric representation of sigma see A237593 and its related sequences.
Equivalently, numbers k with the property that both Dyck paths of the symmetric representation of sigma(k) have an odd number of peaks. - Omar E. Pol, Sep 13 2018

Examples

			Written as an irregular triangle in which the row lengths are the odd numbers, the sequence begins:
    2;
    7,   8,   9;
   16,  17,  18,  19,  20;
   29,  30,  31,  32,  33,  34,  35;
   46,  47,  48,  49,  50,  51,  52,  53,  54;
   67,  68,  69,  70,  71,  72,  73,  74,  75,  76,  77;
   92,  93,  94,  95,  96,  97,  98,  99, 100, 101, 102, 103, 104;
  121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135;
...
Illustration of initial terms:
-----------------------------------------------------------
   k   sigma(k)   Diagram of the symmetry of sigma
-----------------------------------------------------------
                    _         _ _ _             _ _ _ _ _
                  _| |       | | | |           | | | | | |
   2      3      |_ _|       | | | |           | | | | | |
                             | | | |           | | | | | |
                            _|_| | |           | | | | | |
                          _|  _ _|_|           | | | | | |
                  _ _ _ _|  _| |               | | | | | |
   7      8      |_ _ _ _| |_ _|               | | | | | |
   8     15      |_ _ _ _ _|              _ _ _| | | | | |
   9     13      |_ _ _ _ _|             |  _ _ _|_| | | |
                                        _| |    _ _ _|_| |
                                      _|  _|   |  _ _ _ _|
                                  _ _|  _|  _ _| |
                                 |  _ _|  _|    _|
                                 | |     |     |
                  _ _ _ _ _ _ _ _| |  _ _|  _ _|
  16     31      |_ _ _ _ _ _ _ _ _| |  _ _|
  17     18      |_ _ _ _ _ _ _ _ _| | |
  18     39      |_ _ _ _ _ _ _ _ _ _| |
  19     20      |_ _ _ _ _ _ _ _ _ _| |
  20     42      |_ _ _ _ _ _ _ _ _ _ _|
.
For the first nine terms of the sequence we can see in the above diagram that both Dyck path (the smallest and the largest) of the symmetric representation of sigma(k) have a central peak.
Compare with A317304.

Crossrefs

Column 1 gives A130883, n >= 1.
Column 2 gives A033816, n >= 1.
Row sums give the odd-indexed terms of A006002.
Right border gives the positive terms of A014107, also the odd-indexed terms of A000096.
The union of A000217, A317304 and this sequence gives A001477.
Some other sequences related to the central peak or the central valley of the symmetric representation of sigma are A000217, A000384, A007606, A007607, A014105, A014132, A162917, A161983, A317304. See also A317306.
Cf. A000203, A005408, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A249351, A262626.

A317304 Numbers k with the property that both Dyck paths of the symmetric representation of sigma(k) have a central valley.

Original entry on oeis.org

4, 5, 11, 12, 13, 14, 22, 23, 24, 25, 26, 27, 37, 38, 39, 40, 41, 42, 43, 44, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149
Offset: 1

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Author

Omar E. Pol, Aug 27 2018

Keywords

Comments

Also triangle read by rows which gives the even-indexed rows of triangle A014132.
There are no triangular number (A000217) in this sequence.
For more information about the symmetric representation of sigma see A237593 and its related sequences.
Equivalently, numbers k with the property that both Dyck paths of the symmetric representation of sigma(k) have an even number of peaks. - Omar E. Pol, Sep 13 2018

Examples

			Written as an irregular triangle in which the row lengths are the positive even numbers, the sequence begins:
    4,   5;
   11,  12,  13,  14;
   22,  23,  24,  25,  26,  27;
   37,  38,  39,  40,  41,  42,  43,  44;
   56,  57,  58,  59,  60,  61,  62,  63,  64,  65;
   79,  80,  81,  82,  83,  84,  85,  86,  87,  88,  89,  90;
  106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119;
...
Illustration of initial terms:
-------------------------------------------------
   k  sigma(k)  Diagram of the symmetry of sigma
-------------------------------------------------
                       _ _           _ _ _ _
                      | | |         | | | | |
                     _| | |         | | | | |
                 _ _|  _|_|         | | | | |
   4      7     |_ _ _|             | | | | |
   5      6     |_ _ _|             | | | | |
                                 _ _|_| | | |
                               _|    _ _|_| |
                             _|     |  _ _ _|
                            |      _|_|
                 _ _ _ _ _ _|  _ _|
  11     12     |_ _ _ _ _ _| |  _|
  12     28     |_ _ _ _ _ _ _| |
  13     14     |_ _ _ _ _ _ _| |
  14     24     |_ _ _ _ _ _ _ _|
.
For the first six terms of the sequence we can see in the above diagram that both Dyck path (the smallest and the largest) of the symmetric representation of sigma(k) have a central valley.
Compare with A317303.

Crossrefs

Row sums give A084367. n >= 1.
Column 1 gives A084849, n >= 1.
Column 2 gives A096376, n >= 1.
Right border gives the nonzero terms of A014106.
The union of A000217, A317303 and this sequence gives A001477.
Some other sequences related to the central peak or the central valley of the symmetric representation of sigma are A000217, A000384, A007606, A007607, A014105, A014132, A162917, A161983, A317303. See also A317306.
Cf. A000203, A005843, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A249351, A262626.
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