A239931 -id:A239931 - cp's OEIS frontend

A241838 Column 1 of A237270, also the right border.

1, 3, 2, 7, 3, 12, 4, 15, 5, 9, 6, 28, 7, 12, 8, 31, 9, 39, 10, 42, 11, 18, 12, 60, 13, 21, 14, 56, 15, 72, 16, 63, 17, 27, 18, 91, 19, 30, 20, 90, 21, 96, 22, 42, 23, 36, 24, 124, 25, 39, 26, 49, 27, 120, 28, 120, 29, 45, 30, 168, 31, 48, 32, 127

Offset: 1

Omar E. Pol, Apr 29 2014

First differs from A241559 at a(45).
If A237271(n) = 1 then a(n) = A241558(n) = A241559(n) = A000203(n).
If n is an odd prime then a(n) = (n + 1)/2 = A241558(n) = A241559(n).
For more information see A237593.

			For n = 45 the symmetric representation of sigma(45) = 78 has three parts [23, 32, 23], both the first and the last term are equal to 23, so a(45) = 23.

Robert Price, Table of n, a(n) for n = 1..10000

Cf. A000203, A071561, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A241558, A241559, A245092, A262626.

Map[First[a237270[#]]&,Range[64]] (* data : computing all parts *)
(* computing only the first part of the symmetric representation of sigma(n) *)
row[n_] := Floor[(Sqrt[8n+1]-1)/2] (* in A237591 *)
f[n_, k_] := If[Mod[n-k*(k+1)/2, k]==0, (-1)^(k+1), 0]
g[n_, k_] := Ceiling[(n+1)/k-(k+1)/2] - Ceiling[(n+1)/(k+1)-(k+2)/2] (* in A237591 *)
a241838[n_] := Module[{r=row[n], widths={}, i=1, w=0, len, legs}, w+=f[n, i]; While[i<=r && w!=0, AppendTo[widths, w]; i++; w+=f[n, i]]; len=Length[widths]; legs=Map[g[n, #]&, Range[len]]; If[lenHartmut F. W. Hoft, Jan 25 2018 *)

a(n) = A237270(n, 1) = A237270(n, A237271(n)).

A244370 Total number of toothpicks after n-th stage in the toothpick structure of the symmetric representation of sigma in the four quadrants.

Original entry on oeis.org

8, 24, 48, 80, 112, 160, 200, 264, 328, 408, 464, 560, 624, 728, 832, 960, 1040, 1184, 1272, 1432, 1576, 1728, 1832, 2024, 2160, 2336, 2512, 2736

Offset: 1

Omar E. Pol, Jun 26 2014

Partial sums of A244371.
If we use toothpicks of length 1/2, so the area of the central square is equal to 1. The total area of the structure after n-th stage is equal to A024916(n), the sum of all divisors of all positive integers <= n, hence the total area of the n-th set of symmetric regions added at n-th stage is equal to sigma(n) = A000203(n), the sum of divisors of n.
If we use toothpicks of length 1, so the number of cells (and the area) of the central square is equal to 4. The number of cells (and the total area) of the structure after n-th stage is equal to 4*A024916(n) = A243980(n), hence the number of cells (and the total area) of the n-th set of symmetric regions added at n-th stage is equal to 4*A000203(n) = A239050(n).

			Illustration of the structure after 16 stages (Contains 960 toothpicks):
.
.                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.             _ _| |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  | |_ _
.           _|  _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _  |_
.         _|  _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_  |_
.        |  _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_  |
.   _ _ _| |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  | |_ _ _
.  |  _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _  |
.  | | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | | |
.  | | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | | |
.  | | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | | |
.  | | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | | |
.  | | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | | |
.  | | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | | |
.  | | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | | |
.  | | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | | |
.  | | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | | |
.  | | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | | |
.  | | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | | |
.  | |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_| |
.  |_ _ _  | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |  _ _ _|
.        | |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _| |
.        |_  |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|  _|
.          |_  |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|  _|
.            |_ _  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |  _ _|
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.

Cf. A000203, A004125, A024916, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239050, A239660, A239931-A239934, A243980, A244050, A244360-A244363, A244371, A244970, A244971, A245092.

a(n) = 4*A244362(n) = 8*A244360(n).

a(8) corrected and more terms from Omar E. Pol, Oct 18 2014

A241559 Largest part of the symmetric representation of sigma(n).

Original entry on oeis.org

1, 3, 2, 7, 3, 12, 4, 15, 5, 9, 6, 28, 7, 12, 8, 31, 9, 39, 10, 42, 11, 18, 12, 60, 13, 21, 14, 56, 15, 72, 16, 63, 17, 27, 18, 91, 19, 30, 20, 90, 21, 96, 22, 42, 32, 36, 24, 124, 25, 39, 26, 49, 27, 120, 28, 120, 29, 45, 30, 168, 31, 48, 32, 127

Offset: 1

Michel Marcus and Omar E. Pol, Apr 29 2014

nonn

First differs from A241838 at a(45).
If A237271(n) = 1 then a(n) = A241558(n) = A241838(n) = A000203(n).
If n is an odd prime then a(n) = (n + 1)/2 = A241558(n) = A241838(n).
For more information see A237593.

			For n = 9 the symmetric representation of sigma(9) = 13 in the first quadrant looks like this:
y
.
._ _ _ _ _ 5
|_ _ _ _ _|
.         |_ _ 3
.         |_  |
.           |_|_ _ 5
.               | |
.               | |
.               | |
.               | |
. . . . . . . . |_| . . x
.
There are three parts [5, 3, 5] and the largest part is 5 so a(9) = 5.
For n = 45 the symmetric representation of sigma(45) = 78 has three parts [23, 32, 23] and the largest part is 32 so a(45) = 32.

Cf. A000203, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A241558, A071561, A071562, A241838.

(* Function a237270[] is defined in A237270 *)
a241559[n_]:=Max[a237270[n]]
Map[a241559,Range[64]] (* data *)
(* Hartmut F. W. Hoft, Sep 19 2014 *)

A244971 Number of regions in the symmetric representation of sigma(n) on the four quadrants.

Original entry on oeis.org

1, 1, 4, 1, 4, 1, 4, 1, 8, 4, 4, 1, 4, 4, 8, 1, 4, 1, 4, 1, 12, 4, 4, 1, 8, 4, 12, 1, 4, 1, 4, 1, 12, 4, 8, 1, 4, 4, 12, 1, 4, 1, 4, 4, 8, 4, 4, 1, 8, 8, 12, 4, 4, 1, 12, 1, 12, 4, 4, 1, 4, 4, 16, 1, 12, 1, 4, 4, 12, 8, 4, 1, 4, 4, 12, 4, 8, 4, 4, 1, 16, 4, 4, 1, 12, 4, 12, 1, 4, 1

Offset: 1

Omar E. Pol, Jul 08 2014

nonn

Partial sums give A244970.

Number of terraces at the n-th level (starting from the top) of the stepped pyramid described in A244050. - Omar E. Pol, Apr 20 2016

			From _Omar E. Pol_, Apr 20 2016: (Start)
Illustration of the top view of the stepped pyramid with 16 levels:
.                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.             _ _| |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  | |_ _
.           _|  _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _  |_
.         _|  _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_  |_
.        |  _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_  |
.   _ _ _| |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  | |_ _ _
.  |  _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _  |
.  | | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | | |
.  | | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | | |
.  | | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | | |
.  | | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | | |
.  | | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | | |
.  | | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | | |
.  | | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | | |
.  | | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | | |
.  | | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | | |
.  | | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | | |
.  | | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | | |
.  | |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_| |
.  |_ _ _  | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |  _ _ _|
.        | |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _| |
.        |_  |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|  _|
.          |_  |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|  _|
.            |_ _  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |  _ _|
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
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.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..5000 (computed from the b-file of A237271 provided by Michel Marcus)

Cf. A000203, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239050, A239660, A239931-A239934, A244050, A244370, A244371, A244970, A245092.

lista() = {v = readvec("b237271.txt"); for (i=1, #v, vi = v[i]; if (vi == 1, w = 1, w = 4*(vi-1)); print1(w, ", "););} \\ Michel Marcus, Sep 29 2014

a(n) = 1, if A237271(n) = 1.

a(n) = 4*(A237271(n) - 1), if A237271(n) > 1.

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

Omar E. Pol, Sep 28 2018

nonn

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.

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

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.

A279228 Number of unit steps that are shared by the smallest and largest Dyck path of the symmetric representation of sigma(n).

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 4, 0, 2, 0, 8, 0, 10, 2, 4, 0, 14, 0, 16, 0, 6, 6, 20, 0, 16, 8, 10, 0, 26, 0, 28, 0

Offset: 1

Omar E. Pol, Dec 08 2016

a(n) is also the number of unit steps that are shared by the largest Dyck path of the symmetric representation of sigma(n) and the largest Dyck path of the symmetric representation of sigma(n-1), in a quadrant of the square grid.

For more information about the Dyck paths of the symmetric representation of sigma(n) see A237593.

			Illustration of initial terms (n = 1..12) using the spiral described in A239660:
.               _ _ _ _ _ _
.              |  _ _ _ _ _|_ _ _ _ _
.         0   _| |         |_ _ _ _ _|
.           _|_ _|                   |_ _ 2
.       _ _| |      _ _ _ _          |_  |
.      |  _ _|  0 _|  _ _ _|_ _ _      |_|_ _
.      | |      _|   |     |_ _ _|  2      | |
.      | |     |  _ _|           |_ _      | |
.      | |     | |    0 _ _        | |     | |
.      | |     | |     |  _|_ 0    | |     | |
.     _|_|    _|_|    _|_| |_|    _|_|    _|_|    _
.    | |     | |     | |         | |     | |     | |
.    | |     | |     |_|_ _     _| |     | |     | |
.    | |     | |      0|_ _|_ _|  _|     | |     | |
.    | |     |_|_          |_ _ _|0   _ _| |     | |
.    | |         |_                 _|  _ _|     | |
.    |_|_ _     4  |_ _ _ _        |  _|    _ _ _| |
.          |_      |_ _ _ _|_ _ _ _| |  0 _|    _ _|
.            |_            |_ _ _ _ _|  _|     |
.         8    |                       |      _|
.              |_ _ _ _ _ _            |  _ _|
.              |_ _ _ _ _ _|_ _ _ _ _ _| |      0
.                          |_ _ _ _ _ _ _|
.
.
For an illustration of the following examples see the last lap of the above spiral starting in the first quadrant.
For n = 9 the Dyck paths of the symmetric representation of sigma(9) share 2 unit steps, so a(9) = 2.
For n = 10 the Dyck paths of the symmetric representation of sigma(10) meet at the center, but they do not share unit steps, so a(10) = 0.
For n = 11 the Dyck paths of the symmetric representation of sigma(11) share 8 unit steps, so a(11) = 8.
For n = 12 the Dyck paths of the symmetric representation of sigma(12) do not share unit steps, so a(12) = 0.
Note that we can find the spiral on the terraces of the stepped pyramid described in A244050.

Cf. A279029 gives the indices of the zero values.
Cf. A279244 gives the indices of the positive values.
Cf. A000203, A005843, A008586, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A244050, A244361, A244363, A245092, A262626, A348705.

a(n) = 2*n - A244363(n) = 2*(n - A244361(n)).

a(n) = A008586(n) - A348705(n). - Omar E. Pol, Dec 13 2021

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

Omar E. Pol, Aug 27 2018

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

			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.

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

Omar E. Pol, Aug 27 2018

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

			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.

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.

A317305 Sum of divisors of the n-th number whose divisors increase by a factor of 2 or less.

Original entry on oeis.org

1, 3, 7, 12, 15, 28, 31, 39, 42, 60, 56, 72, 63, 91, 90, 96, 124, 120, 120, 168, 127, 144, 195, 186, 224, 180, 234, 252, 217, 210, 280, 248, 360, 312, 255, 336, 336, 403, 372, 392, 378, 363, 480, 372, 546, 508, 399, 468, 465, 504, 434, 576, 600, 504, 504, 560, 546, 744, 728, 511

Offset: 1

Omar E. Pol, Aug 25 2018

nonn

Also consider the n-th number k with the property that the symmetric representation of sigma(k) has only one part. a(n) is the area of the diagram (see the example). For more information see A237593 and its related sequences.

			Illustration of initial terms (n = 1..13):
.
  a(n)
        _ _   _   _   _       _       _   _   _       _       _   _   _
   1   |_| | | | | | | |     | |     | | | | | |     | |     | | | | | |
   3   |_ _|_| | | | | |     | |     | | | | | |     | |     | | | | | |
        _ _|  _|_| | | |     | |     | | | | | |     | |     | | | | | |
   7   |_ _ _|    _|_| |     | |     | | | | | |     | |     | | | | | |
        _ _ _|  _|  _ _|     | |     | | | | | |     | |     | | | | | |
  12   |_ _ _ _|  _|    _ _ _| |     | | | | | |     | |     | | | | | |
        _ _ _ _| |    _|    _ _|     | | | | | |     | |     | | | | | |
  15   |_ _ _ _ _|  _|     |    _ _ _| | | | | |     | |     | | | | | |
                   |      _|   |  _ _ _|_| | | |     | |     | | | | | |
                   |  _ _|    _| |    _ _ _|_| |     | |     | | | | | |
        _ _ _ _ _ _| |      _|  _|   |  _ _ _ _|     | |     | | | | | |
  28   |_ _ _ _ _ _ _|  _ _|  _|  _ _| |    _ _ _ _ _| |     | | | | | |
                       |  _ _|  _|    _|   |    _ _ _ _|     | | | | | |
                       | |     |     |  _ _|   |    _ _ _ _ _| | | | | |
        _ _ _ _ _ _ _ _| |  _ _|  _ _|_|       |   |  _ _ _ _ _|_| | | |
  31   |_ _ _ _ _ _ _ _ _| |  _ _|  _|      _ _|   | |    _ _ _ _ _|_| |
        _ _ _ _ _ _ _ _ _| | |     |      _|    _ _| |   |  _ _ _ _ _ _|
  39   |_ _ _ _ _ _ _ _ _ _| |  _ _|    _|  _ _|  _ _|   | |
        _ _ _ _ _ _ _ _ _ _| | |       |   |    _|    _ _| |
  42   |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|  _|  _|     |  _ _|
                               | |       |  _|      _| |
                               | |  _ _ _| |      _|  _|
        _ _ _ _ _ _ _ _ _ _ _ _| | |  _ _ _|  _ _|  _|
  60   |_ _ _ _ _ _ _ _ _ _ _ _ _| | |       |  _ _|
                                   | |  _ _ _| |
                                   | | |  _ _ _|
        _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | |
  56   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
        _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
  72   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
        _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  63   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
The length of the largest Dyck path of the n-th diagram equals A047836(n).
The semilength equals A174973(n).
a(n) is the area of the n-th diagram.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

A317307 is a subsequence.
Cf. A174973.
Cf. A000203, A047836, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A262626, A361208.

A317305[upto_]:=Table[If[AllTrue[Map[Last[#]/First[#]&,Partition[Divisors[n],2,1]],#<=2&],DivisorSigma[1,n],Nothing],{n,upto}];
A317305[500] (* Paolo Xausa, Jan 12 2023 *)

a(n) = A000203(A174973(n)).

A348705 a(n) is the total length of all line segments in the symmetric representation of sigma(n).

Original entry on oeis.org

4, 8, 12, 16, 18, 24, 24, 32, 34, 40, 36, 48, 42, 54, 56, 64, 54, 72, 60, 80, 78, 82, 72, 96, 84, 96, 98, 112, 90, 120, 96, 128

Offset: 1

Omar E. Pol, Oct 30 2021

a(n) is also the number of toothpicks of length 1 needed to represent the symmetric representation of sigma(n) (see the examples).
The diagram is symmetric thus all terms are even.
If the symmetric representation of sigma(n) has only one part (cf. A174973) or if it has two parts and they meet at the center of the Dyck path (cf. A262259) then a(n) = 4*n, otherwise a(n) < 4*n. In other words: if n is a term of A279029 then a(n) = 4*n, otherwise a(n) < 4*n.

			Illustration of initial terms:
.                                                          _ _ _ _
.                                            _ _ _        |_ _ _  |_
.                                _ _ _      |_ _ _|             |   |_
.                      _ _      |_ _  |_          |_ _          |_ _  |
.              _ _    |_ _|_        |_  |           | |             | |
.        _    |_  |       | |         | |           | |             | |
.       |_|     |_|       |_|         |_|           |_|             |_|
.
n:       1      2        3          4           5               6
a(n):    4      8       12         16          18              24
.
.                                                  _ _ _ _ _
.                            _ _ _ _ _            |_ _ _ _ _|
.        _ _ _ _            |_ _ _ _  |                     |_ _
.       |_ _ _ _|                   | |_                    |_  |
.               |_                  |_  |_ _                  |_|_ _
.                 |_ _                |_ _  |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   |_|                   |_|                     |_|
.
n:              7                    8                      9
a(n):          24                   32                     34
.
Another way for the illustration of initial terms is as follows:
--------------------------------------------------------------------------
.  n  a(n)                             Diagram
--------------------------------------------------------------------------
            _
   1   4   |_|  _
              _| |  _
   2   8     |_ _| | |  _
                _ _|_| | |  _
   3  12       |_ _|  _| | | |  _
                  _ _|  _| | | | |  _
   4  16         |_ _ _|  _|_| | | | |  _
                    _ _ _|  _ _| | | | | |  _
   5  18           |_ _ _| |    _| | | | | | |  _
                      _ _ _|  _|  _|_| | | | | | |  _
   6  24             |_ _ _ _|  _|  _ _| | | | | | | |  _
                        _ _ _ _|  _|  _ _| | | | | | | | |  _
   7  24               |_ _ _ _| |  _|  _ _|_| | | | | | | | |  _
                          _ _ _ _| |  _| |  _ _| | | | | | | | | |  _
   8  32                 |_ _ _ _ _| |_ _| |  _ _| | | | | | | | | | |  _
                            _ _ _ _ _|  _ _|_|  _ _|_| | | | | | | | | | |
   9  34                   |_ _ _ _ _| |  _|  _|  _ _ _| | | | | | | | | |
                              _ _ _ _ _| |  _|  _|    _ _| | | | | | | | |
  10  40                     |_ _ _ _ _ _| |  _|     |  _ _|_| | | | | | |
                                _ _ _ _ _ _| |      _| |  _ _ _| | | | | |
  11  36                       |_ _ _ _ _ _| |  _ _|  _| |  _ _ _| | | | |
                                  _ _ _ _ _ _| |  _ _|  _|_|  _ _ _|_| | |
  12  48                         |_ _ _ _ _ _ _| |  _ _|  _ _| |  _ _ _| |
                                    _ _ _ _ _ _ _| |  _| |    _| |  _ _ _|
  13  42                           |_ _ _ _ _ _ _| | |  _|  _|  _| |
                                      _ _ _ _ _ _ _| | |_ _|  _|  _|
  14  54                             |_ _ _ _ _ _ _ _| |  _ _|  _|
                                        _ _ _ _ _ _ _ _| |  _ _|
  15  56                               |_ _ _ _ _ _ _ _| | |
                                          _ _ _ _ _ _ _ _| |
  16  64                                 |_ _ _ _ _ _ _ _ _|
...

Cf. A008586 (upper bounds).
Cf. A237271 (number of parts or regions).
Cf. A340833 (number of vertices).
Cf. A340846 (number of edges).
Cf. A239931-A239934 (illustration of first 32 diagrams).
Cf. A000203, A139250, A174973, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A238443, A239660, A245092, A262259, A262626, A279029, A279228, A348854.

a(n) = 2*A348854(n).

a(n) = A008586(n) - A279228(n). - Omar E. Pol, Dec 13 2021

cp's OEIS Frontend

A241838 Column 1 of A237270, also the right border.

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A244370 Total number of toothpicks after n-th stage in the toothpick structure of the symmetric representation of sigma in the four quadrants.

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A241559 Largest part of the symmetric representation of sigma(n).

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A244971 Number of regions in the symmetric representation of sigma(n) on the four quadrants.

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A319801 Odd numbers without middle divisors.

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A279228 Number of unit steps that are shared by the smallest and largest Dyck path of the symmetric representation of sigma(n).

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

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

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A317305 Sum of divisors of the n-th number whose divisors increase by a factor of 2 or less.

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