A239934 -id:A239934 - cp's OEIS frontend

A239663 a(n) is the smallest number k such that the symmetric representation of sigma(k) has n parts.

1, 3, 9, 21, 63, 147, 357, 903, 2499, 6069, 13915, 29095, 59455, 142945, 320045, 643885, 1367465, 3287735, 6779135, 13853015, 30262595, 61773745

Offset: 1

Omar E. Pol, Mar 23 2014

Conjecture 1: where records occur in A237271. - Omar E. Pol, Dec 27 2016
For more information about the symmetric representation of sigma see A237270, A237593.
This sequence of (first occurrence of) parts appears to be strictly increasing in contrast to sequence A250070 of (first occurrence of) maximum widths. - Hartmut F. W. Hoft, Dec 09 2014
Conjecture 2: all terms are odd numbers. - Omar E. Pol, Oct 14 2018
Proof of Conjecture 2: Let n = 2^m * q with m>0 and q odd; then the 1's in even positions of row n in the triangle of A237048 are at positions 2^(m+1) * d <= row(n) where d divides q. For n/2 the even positions of 1's occur at the smaller values 2^m * d <= row(n/2), thus either keeping or reducing widths (A249223) of parts in the symmetric representation of sigma for n/2 inherited from row n. Therefore the number of parts for n is at most as large as for n/2, i.e., all numbers in this sequence are odd. - Hartmut F. W. Hoft, Sep 22 2021
Observation: at least for n = 1..21 we have that 2*a(n) < a(n+1). - Omar E. Pol, Sep 22 2021
From Omar E. Pol, Jul 28 2025: (Start)
Conjecture 3: a(n) is the smallest number k having n 2-dense sublists of divisors of k.
The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.
In a sublist of divisors of k the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of k.
An example of the conjecture 3 for n = 1..5 is as shown below:
  ----------------------------------------------------
  |    |  List of divisors of k         |     |      |
  |  k |  [with sublists in brackets]   |  n  | a(n) |
  ----------------------------------------------------
  |  1 |  [1];                          |  1  |   1  |
  |  3 |  [1], [3];                     |  2  |   3  |
  |  9 |  [1], [3], [9];                |  3  |   9  |
  | 21 |  [1], [3], [7], [21];          |  4  |  21  |
  | 63 |  [1], [3], [7, 9], [21], [63]; |  5  |  63  |
(End)
Conjecture 4: a(n) is the smallest number k having n divisors p of k such that p is greater than twice the adjacent previous divisor of k. - Omar E. Pol, Aug 05 2025

			------------------------------------------------------
n       a(n)     A239665                  A266094(n)
------------------------------------------------------
1        1       [1]                           1
2        3       [2, 2]                        4
3        9       [5, 3, 5]                    13
4       21       [11, 5, 5, 11]               32
5       63       [32, 12, 16, 12, 32]        104
...
For n = 3 the symmetric representation of sigma(9) = 13 contains three parts [5, 3, 5] as shown below:
.
.     _ _ _ _ _ 5
.    |_ _ _ _ _|
.              |_ _ 3
.              |_  |
.                |_|_ _ 5
.                    | |
.                    | |
.                    | |
.                    | |
.                    |_|
.

Hartmut F. W. Hoft, Procedural implementation for extension values
Omar E. Pol, Perspective view of the stepped pyramid (16 levels)

Row 1 of A240062.
Cf. A000203, A196020, A236104, A235791, A237048, A237270, A237271, A237591, A237593, A238443, A239657, A239660, A239665, A239931-A239934, A245092, A262626, A266094.
Cf. A249223, A250070, A262045, A320521, A341969, A341970, A341971, A347980.
Cf. A174973 (2-dense numbers), A384149, A384222, A384225, A384226.

(* a239663[] permits computation in intervals *)
(* Function a237270[] is defined in A237270 *)
(* variable "list" contains the first occurrences up to m *)
a239663[list_,{m_, n_}]:=Module[{firsts=list, g=Length[list], i, p}, For[i=m, i<=n, i++, p=Length[a237270[i]]; If[p>g, AppendTo[firsts, i]; g=p]]; firsts]
a239663[{1}, {1, 1000}] (* computes the first 8 values *)
(* Hartmut F. W. Hoft, Jul 08 2014 *)
(* support functions are defined in A341969, A341970 & A341971 *)
a239663[n_, len_] := Module[{list=Table[0, len], i, v}, For[i=1, i<=n, i+=2, v=Count[a341969[i], 0]+1;If[list[[v]]==0, list[[v]]=i]]; list]
a239663[62000000,22] (* Hartmut F. W. Hoft, Sep 22 2021 *)

a(6)-a(8) from Michel Marcus, Mar 28 2014
a(9) from Michel Marcus, Mar 29 2014
a(10)-a(11) from Michel Marcus, Apr 02 2014
a(12) from Hartmut F. W. Hoft, Jul 08 2014
a(13)-a(18) from Hartmut F. W. Hoft, Dec 09 2014
a(19)-a(22) from Hartmut F. W. Hoft, Sep 22 2021

A240062 Square array read by antidiagonals in which T(n,k) is the n-th number j with the property that the symmetric representation of sigma(j) has k parts, with j >= 1, n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 6, 7, 15, 21, 8, 10, 25, 27, 63, 12, 11, 35, 33, 81, 147, 16, 13, 45, 39, 99, 171, 357, 18, 14, 49, 51, 117, 189, 399, 903, 20, 17, 50, 55, 153, 207, 441, 987, 2499, 24, 19, 70, 57, 165, 243, 483, 1029, 2709, 6069, 28, 22, 77, 65, 195, 261, 513, 1113

Offset: 1

Omar E. Pol, Apr 06 2014

This is a permutation of the positive integers.
All odd primes are in column 2 (together with some even composite numbers) because the symmetric representation of sigma(prime(i)) is [m, m], where m = (1 + prime(i))/2, for i >= 2.
The union of all odd-indexed columns gives A071562, the positive integers that have middle divisors. The union of all even-indexed columns gives A071561, the positive integers without middle divisors. - Omar E. Pol, Oct 01 2018
Each column in the table of A357581 is a subsequence of the respective column in the table of this sequence; however, the first row in the table of A357581 is not a subsequence of the first row in the table of this sequence. - Hartmut F. W. Hoft, Oct 04 2022
Conjecture: T(n,k) is the n-th positive integer with k 2-dense sublists of divisors. - Omar E. Pol, Aug 25 2025

			Array begins:
   1,  3,  9, 21,  63, 147, 357,  903, 2499, 6069, ...
   2,  5, 15, 27,  81, 171, 399,  987, 2709, 6321, ...
   4,  7, 25, 33,  99, 189, 441, 1029, 2793, 6325, ...
   6, 10, 35, 39, 117, 207, 483, 1113, 2961, 6783, ...
   8, 11, 45, 51, 153, 243, 513, 1197, 3025, 6875, ...
  12, 13, 49, 55, 165, 261, 567, 1239, 3087, 6909, ...
  16, 14, 50, 57, 195, 275, 609, 1265, 3249, 7011, ...
  18, 17, 70, 65, 231, 279, 621, 1281, 3339, 7203, ...
  20, 19, 77, 69, 255, 297, 651, 1375, 3381, 7353, ...
  24, 22, 91, 75, 273, 333, 729, 1407, 3591, 7581, ...
  ...
[Lower right hand triangle of array completed by _Hartmut F. W. Hoft_, Oct 04 2022]

Index entries for sequences that are permutations of the natural numbers

For programs see A237271 and A237593.
Row 1 is A239663.
Column 1 is A174973.
Columns 2..7: A239929, A279102, A280107, A320066, A320511, A357775.
For more information see A239663 and A239665.
Cf. A000203, A006254, A065091, A067742, A071561, A071562, A196020, A236104, A235791, A237048, A237270, A239660, A239929, A239931-A239934, A245092, A262626, A319529, A319796, A319801, A319802.
Cf. A341969, A341970, A341971, A357581.

(* function a341969 and support functions are defined in A341969, A341970 and A341971 *)
partsSRS[n_] := Length[Select[SplitBy[a341969[n], #!=0&], #[[1]]!=0&]]
widthTable[n_, {r_, c_}] := Module[{k, list=Table[{}, c], parts}, For[k=1, k<=n, k++, parts=partsSRS[k]; If[parts<=c&&Length[list[[parts]]]=1, j--, vec[[PolygonalNumber[i+j-2]+j]]=arr[[i, j]]]]; vec]
a240062T[n_, r_] := TableForm[widthTable[n, {r, r}]]
a240062[6069, 10] (* data *)
a240062T[7581, 10] (* 10 X 10 array - Hartmut F. W. Hoft, Oct 04 2022 *)

a(n) > 128 from Michel Marcus, Apr 08 2014

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

Omar E. Pol, Mar 29 2014

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

			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.

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

A239933 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(4n-1).

Original entry on oeis.org

2, 2, 4, 4, 6, 6, 8, 8, 8, 10, 10, 12, 12, 14, 6, 6, 14, 16, 16, 18, 12, 18, 20, 8, 8, 20, 22, 22, 24, 24, 26, 10, 10, 26, 28, 8, 8, 28, 30, 30, 32, 12, 16, 12, 32, 34, 34, 36, 36, 38, 24, 24, 38, 40, 40, 42, 42, 44, 16, 16, 44, 46, 20, 46, 48, 12, 12, 48, 50, 18, 20, 18, 50, 52, 52, 54, 54, 56, 20, 20, 56, 58, 14, 14, 58, 60, 12, 12, 60, 62, 22, 22, 62, 64, 64

Offset: 1

Omar E. Pol, Mar 29 2014

Row n is a palindromic composition of sigma(4n-1).
Row n is also the row 4n-1 of A237270.
Row n has length A237271(4n-1).
Row sums give A239053.
Also row n lists the parts of the symmetric representation of sigma in the n-th arm of the third 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), see A239934.
We can find the spiral (mentioned above) on the terraces of the pyramid described in A244050. - Omar E. Pol, Dec 06 2016

			The irregular triangle begins:
2, 2;
4, 4;
6, 6;
8, 8, 8;
10, 10;
12, 12;
14, 6, 6, 14;
16, 16;
18, 12, 18;
20, 8, 8, 20;
22, 22;
24, 24;
26, 10, 10, 26;
28, 8, 8, 28;
30, 30;
32, 12, 16, 12, 32;
...
Illustration of initial terms in the third quadrant of the spiral described in A239660:
.     _       _       _       _       _       _       _       _
.    | |     | |     | |     | |     | |     | |     | |     | |
.    | |     | |     | |     | |     | |     | |     | |     |_|_ _
.    | |     | |     | |     | |     | |     | |     | |    2  |_ _|
.    | |     | |     | |     | |     | |     | |     |_|_     2
.    | |     | |     | |     | |     | |     | |    4    |_
.    | |     | |     | |     | |     | |     |_|_ _        |_ _ _ _
.    | |     | |     | |     | |     | |    6      |_      |_ _ _ _|
.    | |     | |     | |     | |     |_|_ _ _        |_   4
.    | |     | |     | |     | |    8      | |_ _      |
.    | |     | |     | |     |_|_ _ _      |_    |     |_ _ _ _ _ _
.    | |     | |     | |   10        |       |_  |_    |_ _ _ _ _ _|
.    | |     | |     |_|_ _ _ _      |_ _   8  |_ _|  6
.    | |     | |   12          |         |_        |
.    | |     |_|_ _ _ _ _      |_ _        |       |_ _ _ _ _ _ _ _
.    | |   14          | |         |_      |_ _    |_ _ _ _ _ _ _ _|
.    |_|_ _ _ _ _      | |_ _        |_        |  8
.  16            |     |_ _  |         |       |
.                |         |_|_        |_ _    |_ _ _ _ _ _ _ _ _ _
.                |_ _     6    |_ _        |   |_ _ _ _ _ _ _ _ _ _|
.                    |         |_  |       | 10
.                    |_       6  | |_ _    |
.                      |_        |_ _ _|   |_ _ _ _ _ _ _ _ _ _ _ _
.                        |_ _          |   |_ _ _ _ _ _ _ _ _ _ _ _|
.                            |         | 12
.                            |_ _ _    |
.                                  |   |_ _ _ _ _ _ _ _ _ _ _ _ _ _
.                                  |   |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
.                                  | 14
.                                  |
.                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.                                16
.
For n = 7 we have that 4*7-1 = 27 and the 27th row of A237593 is [14, 5, 3, 2, 1, 2, 2, 1, 2, 3, 5, 14] and the 26th row of A237593 is [14, 5, 2, 2, 2, 1, 1, 2, 2, 2, 5, 14] therefore between both Dyck paths there are four regions (or parts) of sizes [14, 6, 6, 14], so row 7 is [14, 6, 6, 14].
The sum of divisors of 27 is 1 + 3 + 9 + 27 = A000203(27) = 40. On the other hand the sum of the parts of the symmetric representation of sigma(27) is 14 + 6 + 6 + 14 = 40, equaling the sum of divisors of 27.

Cf. A000203, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A239053, A239660, A239931, A239932, A239934, A244050, A245092, A262626.

A237590 a(n) is the total number of regions (or parts) after n-th stage in the diagram of the symmetries of sigma described in A236104.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 11, 14, 16, 18, 19, 21, 23, 26, 27, 29, 30, 32, 33, 37, 39, 41, 42, 45, 47, 51, 52, 54, 55, 57, 58, 62, 64, 67, 68, 70, 72, 76, 77, 79, 80, 82, 84, 87, 89, 91, 92, 95, 98, 102, 104, 106, 107, 111, 112, 116, 118, 120, 121, 123, 125, 130, 131, 135, 136, 138, 140, 144, 147, 149, 150, 152, 154

Offset: 1

Omar E. Pol, Mar 31 2014

nonn

The total area (or total number of cells) of the diagram after n stages is equal to A024916(n), the sum of all divisors of all positive integers <= n.
Note that the region between the virtual circumscribed square and the diagram is a symmetric polygon whose area is equal to A004125(n), see example.
For more information see A237593 and A237270.
a(n) is also the total number of terraces of the stepped pyramid with n levels described in A245092. - Omar E. Pol, Apr 20 2016

			Illustration of initial terms:
.                                                         _ _ _ _
.                                           _ _ _        |_ _ _  |_
.                               _ _ _      |_ _ _|       |_ _ _|   |_
.                     _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
.             _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
.       _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
.      |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
.
.
.       1      2        4          5            7              8
.
For n = 6 the diagram contains 8 regions (or parts), so a(6) = 8.
The sum of all divisors of all positive integers <= 6 is [1] + [1+2] + [1+3] + [1+2+4] + [1+5] + [1+2+3+6] = 33. On the other hand after 6 stages the sum of all parts of the diagram is [1] + [3] + [2+2] + [7] + [3+3] + [12] = 33, equaling the sum of all divisors of all positive integers <= 6.
Note that the region between the virtual circumscribed square and the diagram is a symmetric polygon whose area is equal to A004125(6) = 3.
From _Omar E. Pol_, Dec 25 2020: (Start)
Illustration of the diagram after 29 stages (contain 215 vertices, 268 edges and 54 regions or parts):
._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
|_ _ _ _ _ _ _ _ _ _ _ _ _ _  |
|_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
|_ _ _ _ _ _ _ _ _ _ _ _ _  | |
|_ _ _ _ _ _ _ _ _ _ _ _ _| | |
|_ _ _ _ _ _ _ _ _ _ _ _  | | |_ _ _
|_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _  |
|_ _ _ _ _ _ _ _ _ _ _  | | |_ _  | |_
|_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_  |_
|_ _ _ _ _ _ _ _ _ _  | |       |_ _|   |_
|_ _ _ _ _ _ _ _ _ _| | |_ _    |_  |_ _  |_ _
|_ _ _ _ _ _ _ _ _  | |_ _ _|     |_  | |_ _  |
|_ _ _ _ _ _ _ _ _| | |_ _  |_      |_|_ _  | |
|_ _ _ _ _ _ _ _  | |_ _  |_ _|_        | | | |_ _ _ _ _ _
|_ _ _ _ _ _ _ _| |     |     | |_ _    | |_|_ _ _ _ _  | |
|_ _ _ _ _ _ _  | |_ _  |_    |_  | |   |_ _ _ _ _  | | | |
|_ _ _ _ _ _ _| |_ _  |_  |_ _  | | |_ _ _ _ _  | | | | | |
|_ _ _ _ _ _  | |_  |_  |_    | |_|_ _ _ _  | | | | | | | |
|_ _ _ _ _ _| |_ _|   |_  |   |_ _ _ _  | | | | | | | | | |
|_ _ _ _ _  |     |_ _  | |_ _ _ _  | | | | | | | | | | | |
|_ _ _ _ _| |_      | |_|_ _ _  | | | | | | | | | | | | | |
|_ _ _ _  |_ _|_    |_ _ _  | | | | | | | | | | | | | | | |
|_ _ _ _| |_  | |_ _ _  | | | | | | | | | | | | | | | | | |
|_ _ _  |_  |_|_ _  | | | | | | | | | | | | | | | | | | | |
|_ _ _|   |_ _  | | | | | | | | | | | | | | | | | | | | | |
|_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | | |
|_ _|_  | | | | | | | | | | | | | | | | | | | | | | | | | |
|_  | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
(End)

Robert Price, Table of n, a(n) for n = 1..5000
Omar E. Pol, An infinite stepped pyramid
Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Perspective view of the stepped pyramid (first 16 levels)

Partial sums of A237271.
Compare with A060831 (analog for the diagram that contains subparts).
Cf. A000203, A004125, A024916, A196020, A236104, A235791, A237048, A237270, A237591, A237593, A239659, A239660, A239663, A239665, A239931-A239934, A245092, A244050, A244970, A262626, A317109.

(* total number of parts in the first n symmetric representations *)
(* Function a237270[] is defined in A237270 *)
(* variable "previous" represents the sum from 1 through m-1 *)
a237590[previous_,{m_,n_}]:=Rest[FoldList[Plus[#1,Length[a237270[#2]]]&,previous,Range[m,n]]]
a237590[n_]:=a237590[0,{1,n}]
a237590[78] (* data *)
(* Hartmut F. W. Hoft, Jul 07 2014 *)

a(n) = A317109(n) - A294723(n) + 1 (Euler's formula). - Omar E. Pol, Jul 21 2018

Definition clarified by Omar E. Pol, Jul 21 2018

A319796 Even numbers that have middle divisors, where "middle divisor" means a divisor in the half-open interval [sqrt(n/2), sqrt(n*2)).

Original entry on oeis.org

2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 50, 54, 56, 60, 64, 66, 70, 72, 80, 84, 88, 90, 96, 98, 100, 104, 108, 110, 112, 120, 126, 128, 130, 132, 140, 144, 150, 154, 156, 160, 162, 168, 170, 176, 180, 182, 190, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 238, 240, 242, 252, 256

Offset: 1

Omar E. Pol, Sep 28 2018

nonn

Even numbers k such that the symmetric representation of sigma(k) has an odd number of parts.
An even number A005843 is in this sequence iff A067742(t) != 0.
For the definition of middle divisors, see A067742.
For more information about the symmetric representation of sigma(k) see A237593.
From Hartmut F. W. Hoft, Mar 28 2023: (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)), SRS(a(n)), on the diagonal which equals the triangle entry A249223(n, A003056(n)). The maximum widths of the center part of SRS(a(n)) need not occur at the diagonal.
For example, a(7) = 2 * 3^2 = 18, SRS(18) has a single part with maximum width 2 while its width at the diagonal equals 1 = A067742(18), and divisor 3 is the only middle divisor of a(7). (End)

			6 is in the sequence because it's an even number and the symmetric representation of sigma(6) = 12 has an odd number of parts (more exactly only one part), as shown below:
.    _ _ _ _
.   |_ _ _  |_ 12
.         |   |_
.         |_ _  |
.             | |
.             | |
.             |_|
.
Also 50 is in the sequence because it's an even number and the symmetric representation of sigma(50) = 93 has an odd number of parts (more exactly three parts), they are [39, 15, 39].
a(34) = 110 = 2 * 5 * 11 has 10 and 11 as its middle divisors, and SRS(a(34)) has 3 parts and width 2 at the diagonal. -  _Hartmut F. W. Hoft_, Mar 28 2023

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Intersection of A005843 and A071562.

Cf. A000203, A067742, A071090, A236104, A237270, A237271, A237593, A239932, A239934, A240542, A245092, A280919, A281007, A296508, A299761, A299777, A303297, A319529, A319801, A319802.

filter:= n -> ormap(t -> t^2 >= n/2 and t^2 < 2*n, numtheory:-divisors(n)):
select(filter, 2*[$1..1000]); # Robert Israel, Mar 29 2023

middleDiv[n_] := Select[Divisors[n], Sqrt[n/2]<=#Hartmut F. W. Hoft, Mar 28 2023 *)

Name clarified by Omar E. Pol, Mar 28 2023

A239665 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma of the smallest number whose symmetric representation of sigma has n parts.

Original entry on oeis.org

1, 2, 2, 5, 3, 5, 11, 5, 5, 11, 32, 12, 16, 12, 32, 74, 26, 14, 14, 26, 74, 179, 61, 29, 38, 29, 61, 179, 452, 152, 68, 32, 32, 68, 152, 452, 1250, 418, 182, 152, 100, 152, 182, 418, 1250, 3035, 1013, 437, 342, 85, 85, 342, 437, 1013, 3035, 6958, 1394, 638, 314, 154, 236, 154, 314, 638, 1394, 6958

Offset: 1

Omar E. Pol, Mar 23 2014

Row n is also row A239663(n) of A237270.

			----------------------------------------------------------------------
n    A239663(n)  Triangle begins:                        A266094(n)
----------------------------------------------------------------------
1        1       [1]                                         1
2        3       [2, 2]                                      4
3        9       [5, 3, 5]                                  13
4       21       [11, 5, 5, 11]                             32
5       63       [32, 12, 16, 12, 32]                      104
6      147       [74, 26, 14, 14, 26, 74]                  228
7      357       [179, 61, 29, 38, 29, 61, 179]            576
8      903       [452, 152, 68, 32, 32, 68, 152, 452]     1408
...
Illustration of initial terms:
.
.     _ _ _ _ _ 5
.    |_ _ _ _ _|
.              |_ _ 3
.              |_  |
.                |_|_ _ 5
.                    | |
.     _ _ 2          | |
.    |_ _|_ 2        | |
.     _ 1| |         | |
.    |_| |_|         |_|
.
For n = 2 we have that A239663(2) = 3 is the smallest number whose symmetric representation of sigma has 2 parts. Row 3 of A237593 is [2, 1, 1, 2] and row 2 of A237593 is [2, 2] therefore between both Dyck paths in the first quadrant there are two regions (or parts) of sizes [2, 2], so row 2 is [2, 2].
For n = 3 we have that A239663(3) = 9 is the smallest number whose symmetric representation of sigma has 3 parts. 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 Dyck paths in the first quadrant there are three regions (or parts) of sizes [5, 3, 5], so row 3 is [5, 3, 5].

Cf. A000203, A005279, A196020, A236104, A237270, A237271, A235791, A237591, A237593, A239660, A239663, A239931-A239934, A240020, A240062, A244050, A245092, A262626, A266094.

a(16)-a(28) from Michel Marcus and Omar E. Pol, Mar 28 2014
a(29)-a(36) from Michel Marcus, Mar 28 2014
a(37)-a(45) from Michel Marcus, Mar 29 2014
a(46)-a(66) from Michel Marcus, Apr 02 2014

A241558 Smallest part of the symmetric representation of sigma(n).

Original entry on oeis.org

1, 3, 2, 7, 3, 12, 4, 15, 3, 9, 6, 28, 7, 12, 8, 31, 9, 39, 10, 42, 5, 18, 12, 60, 5, 21, 6, 56, 15, 72, 16, 63, 7, 27, 12, 91, 19, 30, 8, 90, 21, 96, 22, 42, 23, 36, 24, 124, 7, 15, 10, 49, 27, 120, 8, 120, 11, 45, 30, 168, 31, 48, 12, 127, 9, 144, 34, 63, 13

Offset: 1

Michel Marcus and Omar E. Pol, Apr 29 2014

nonn

If A237271(n) = 1 then a(n) = A241559(n) = A241838(n) = A000203(n).
If n is an odd prime then a(n) = (n + 1)/2 = A241559(n) = A241838(n).
For more information see A237270 and 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 smallest part is 3 so a(9) = 3.
For n = 45 the symmetric representation of sigma(45) = 78 has three parts [23, 32, 23] and the smallest part is 23 so a(45) = 23.
For n = 63 the symmetric representation of sigma(63) = 104 has five parts [32, 12, 16, 12, 32] and the smallest part is 12 so a(63) = 12.

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A000203, A071561, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A241559, A241838, A245092, A262626.

(* Function a237270[] is defined in A237270 *)
a241558[n_]:=Min[a237270[n]]
Map[a241558,Range[64]] (* data *)
(* Hartmut F. W. Hoft, Sep 19 2014 *)

More terms from Jinyuan Wang, Feb 14 2020

A319802 Even numbers without middle divisors.

Original entry on oeis.org

10, 14, 22, 26, 34, 38, 44, 46, 52, 58, 62, 68, 74, 76, 78, 82, 86, 92, 94, 102, 106, 114, 116, 118, 122, 124, 134, 136, 138, 142, 146, 148, 152, 158, 164, 166, 172, 174, 178, 184, 186, 188, 194, 202, 206, 212, 214, 218, 222, 226, 230, 232, 236, 244, 246, 248, 250, 254, 258, 262, 268, 274, 278, 282, 284

Offset: 1

Omar E. Pol, Sep 28 2018

nonn

Even numbers k such that the symmetric representation of sigma(k) has an even number of parts.
For the definition of middle divisors, see A067742.
For more information about the symmetric representation of sigma(k) see A237593.
Let p be a prime > 5. Then a(n) is a number of the form m*p where m is an even number < sqrt(p). - David A. Corneth, Sep 28 2018
First differs from A244894 at a(51) = 230. - R. J. Mathar, Oct 04 2018
Is this twice A101550? - Omar E. Pol, Oct 04 2018
This sequence is not twice A101550: first differs at a(57) = 250 != 254 = 2*A101550(57). - Michael S. Branicky, Oct 14 2021

			10 is in the sequence because it's an even number and the symmetric representation of sigma(10) = 18 has an even number of parts as shown below:
.
.     _ _ _ _ _ _ 9
.    |_ _ _ _ _  |
.              | |_
.              |_ _|_
.                  | |_ _ 9
.                  |_ _  |
.                      | |
.                      | |
.                      | |
.                      | |
.                      |_|
.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Intersection of A005843 and A071561.
Cf. A244894 (a subsequence).
Cf. A000203, A067742, A071090, A071562, A236104, A237270, A237271, A237593, A239932, A239934, A240542, A245092, A280919, A281007, A296508, A299761, A299777, A303297, A319529, A319796, A319801.

from sympy import divisors
def ok(n):
    if n < 2 or n%2 == 1: return False
    return not any(n//2 <= d*d < 2*n for d in divisors(n, generator=True))
print(list(filter(ok, range(285)))) # Michael S. Branicky, Oct 14 2021

A243980 Four times the sum of all divisors of all positive integers <= n.

Original entry on oeis.org

4, 16, 32, 60, 84, 132, 164, 224, 276, 348, 396, 508, 564, 660, 756, 880, 952, 1108, 1188, 1356, 1484, 1628, 1724, 1964, 2088, 2256, 2416, 2640, 2760, 3048, 3176, 3428, 3620, 3836, 4028, 4392, 4544, 4784, 5008, 5368, 5536, 5920, 6096, 6432, 6744, 7032, 7224, 7720

Offset: 1

Omar E. Pol, Jun 18 2014

nonn

Also number of "ON" cells at n-th stage in a structure which looks like a simple 2-dimensional cellular automaton (see example). The structure is formed by the reflection on the four quadrants from the diagram of the symmetry of sigma in the first quadrant after n-th stage, hence the area in each quadrant equals the area of each wedge and equals A024916(n); the sum of all divisors of all positive integers <= n. For more information about the diagram see A237593 and A237270.

			Illustration of the structure after 16 stages (contains 880 ON cells):
.                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.             _ _| |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  | |_ _
.           _|  _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _  |_
.         _|  _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_  |_
.        |  _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_  |
.   _ _ _| |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  | |_ _ _
.  |  _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _  |
.  | | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | | |
.  | | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | | |
.  | | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | | |
.  | | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | | |
.  | | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | | |
.  | | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | | |
.  | | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | | |
.  | | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | | |
.  | | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | | |
.  | | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | | |
.  | | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | | |
.  | |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_| |
.  |_ _ _  | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |  _ _ _|
.        | |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _| |
.        |_  |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|  _|
.          |_  |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|  _|
.            |_ _  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |  _ _|
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.

Indranil Ghosh, Table of n, a(n) for n = 1..7342

Partial sums of A239050.
Partial sums give A244050.
Cf. A000203, A000290, A004125, A016742, A024916, A175254, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934.

Accumulate[4*DivisorSigma[1,Range[50]]] (* Harvey P. Dale, May 13 2018 *)

from math import isqrt
def A243980(n): return -(s:=isqrt(n))**2*(s+1) + sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))<<1 # Chai Wah Wu, Oct 21 2023

a(n) = A016742(n) - 4*A004125(n) = 4*A024916(n).

a(n) = 2*(A006218(n) + A222548(n)) = 2*A327329(n). - Omar E. Pol, Sep 25 2019

cp's OEIS Frontend

A239663 a(n) is the smallest number k such that the symmetric representation of sigma(k) has n parts.

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A240062 Square array read by antidiagonals in which T(n,k) is the n-th number j with the property that the symmetric representation of sigma(j) has k parts, with j >= 1, n >= 1, k >= 1.

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

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A239933 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(4n-1).

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A237590 a(n) is the total number of regions (or parts) after n-th stage in the diagram of the symmetries of sigma described in A236104.

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A319796 Even numbers that have middle divisors, where "middle divisor" means a divisor in the half-open interval [sqrt(n/2), sqrt(n*2)).

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A239665 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma of the smallest number whose symmetric representation of sigma has n parts.

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A241558 Smallest part of the symmetric representation of sigma(n).

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