A338156 -id:A338156 - cp's OEIS frontend

A345023 a(n) is the surface area of the symmetric tower described in A221529 which is a polycube whose successive terraces are the symmetric representation of sigma A000203(i) (from i = 1 to n) starting from the top and the levels of these terraces are the partition numbers A000041(h-1) (from h = 1 to n) starting from the base.

6, 16, 32, 58, 90, 142, 202, 292, 406, 562, 754, 1034, 1370, 1822, 2410, 3176, 4136, 5402, 6982, 9026, 11598, 14838, 18894, 24034, 30396, 38312, 48136, 60288, 75220, 93624, 116104, 143598, 177090, 217770, 267106, 326820, 398804, 485472, 589644, 714564, 864000, 1042524, 1255308

Offset: 1

Omar E. Pol, Jun 05 2021

nonn

The largest side of the base of the tower has length n.
The base of the tower is the symmetric representation of A024916(n).
The volume of the tower is equal to A066186(n).
The area of each lateral view of the tower is equal to A000070(n-1).
The growth of the volume of the tower represents the convolution of A000203 and A000041.
The above results are because the correspondence between divisors and partitions described in A338156 and A336812.
The tower is also a member of the family of the stepped pyramid described in A245092.
The equivalent sequence for the surface area of the stepped pyramid is A328366.

			For n = 7 we can see below some views of two associated polycubes called "prism of partitions" and "tower". Both objects contains the same number of cubes (that property is also valid for n >= 1).
     _ _ _ _ _ _ _
    |_ _ _ _      |                 7
    |_ _ _ _|_    |           4     3
    |_ _ _    |   |             5   2
    |_ _ _|_ _|_  |         3   2   2                                    _
    |_ _ _      | |               6 1                 1                 | |
    |_ _ _|_    | |         3     3 1                 1                 | |
    |_ _    |   | |           4   2 1                 1                 | |
    |_ _|_ _|_  | |       2   2   2 1                 1                _|_|
    |_ _ _    | | |             5 1 1               1 1               |   |
    |_ _ _|_  | | |         3   2 1 1               1 1              _|_ _|
    |_ _    | | | |           4 1 1 1             1 1 1             | |   |
    |_ _|_  | | | |       2   2 1 1 1             1 1 1            _|_|_ _|
    |_ _  | | | | |         3 1 1 1 1           1 1 1 1          _| |_ _ _|
    |_  | | | | | |       2 1 1 1 1 1         1 1 1 1 1      _ _|_ _|_ _ _|
    |_|_|_|_|_|_|_|     1 1 1 1 1 1 1     1 1 1 1 1 1 1     |_ _|_|_ _ _ _|
.
       Figure 1.           Figure 2.        Figure 3.           Figure 4.
   Front view of the      Partitions        Position          Lateral view
  prism of partitions.       of 7.         of the 1's.        of the tower.
.
.
                                                             _ _ _ _ _ _ _
                                                            |   | | | | |_|  1
                                                            |   | | |_|_ _|  2
                                                            |   |_|_  |_ _|  3
                                                            |_ _    |_ _ _|  4
                                                                |_  |_ _ _|  5
                                                                  |       |  6
                                                                  |_ _ _ _|  7
.
                                                               Figure 5.
                                                               Top view
                                                             of the tower.
.
Figure 1 is a two-dimensional diagram of the partitions of 7. The area of the diagram is A066186(7) = 105. Note that the diagram can be interpreted also as the front view of a right prism whose volumen is 1*7*A000041(7) = 1*7*15 = 105, equaling the volume of the tower that appears in the figures 4 and 5.
Figure 2 shows the partitions of 7 in accordance with the diagram.
Note that the shape and the area of the lateral view of the tower are the same as the shape and the area where the 1's are located in the diagram of partitions, see the figures 3 and 4. In this case the mentioned area equals A000070(7-1) = 30.
The connection between these two objects is a representation of the correspondence divisor/part described in A338156. See also A336812.

Cf. A000041, A000070, A000203, A024916, A066186, A176206, A196020, A221529, A236104, A237593, A244050, A245092, A327329, A328366, A336811, A336812, A338156.

Accumulate @ Table[4 * PartitionsP[k-1] + 2 * DivisorSigma[1, k], {k, 1, 50}] (* Amiram Eldar, Jul 14 2021 *)

a(n) = 4*A000070(n-1) + 2*A024916(n).

a(n) = 4*A000070(n-1) + A327329(n).

A340057 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the block m consists of the divisors of m multiplied by A000041(n-m), with 1 <= m <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 1, 3, 3, 2, 4, 1, 3, 1, 2, 4, 5, 3, 6, 2, 6, 1, 2, 4, 1, 5, 7, 5, 10, 3, 9, 2, 4, 8, 1, 5, 1, 2, 3, 6, 11, 7, 14, 5, 15, 3, 6, 12, 2, 10, 1, 2, 3, 6, 1, 7, 15, 11, 22, 7, 21, 5, 10, 20, 3, 15, 2, 4, 6, 12, 1, 7, 1, 2, 4, 8, 22, 15, 30, 11, 33, 7, 14, 28, 5, 25

Offset: 1

Omar E. Pol, Dec 27 2020

This triangle is a condensed version of the more irregular triangle A340035.

For further information about the correspondence divisor/part see A338156.

			Triangle begins:
  [1];
  [1],  [1, 2];
  [2],  [1, 2],  [1, 3];
  [3],  [2, 4],  [1, 3],  [1, 2, 4];
  [5],  [3, 6],  [2, 6],  [1, 2, 4],  [1, 5];
  [7],  [5, 10], [3, 9],  [2, 4, 8],  [1, 5],  [1, 2, 3, 6];
  [11], [7, 14], [5, 15], [3, 6, 12], [2, 10], [1, 2, 3, 6], [1, 7];
  ...
Row sums gives A066186.
Written as a tetrahedrons the first five slices are:
  --
  1;
  --
  1,
  1, 2;
  -----
  2,
  1, 2,
  1, 3;
  -----
  3,
  2, 4,
  1, 3,
  1, 2, 4;
  --------
  5,
  3, 6,
  2, 6,
  1, 2, 4,
  1, 5;
  --------
Row sums give A221529.
The slices of the tetrahedron appear in the upper zone of the following table (formed by four zones) which shows the correspondence between divisors and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
|   |    -    |     |       |         |           |  5          |
| C |    -    |     |       |         |  3        |  3 6        |
| O |    -    |     |       |  2      |  2 4      |  2   6      |
| N | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |
| D | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
| D | A027750 |     |       |         |           |  1          |
| I |---------|-----|-------|---------|-----------|-------------|
| V | A027750 |     |       |         |  1        |  1 2        |
| I | A027750 |     |       |         |  1        |  1 2        |
| S | A027750 |     |       |         |  1        |  1 2        |
| O |---------|-----|-------|---------|-----------|-------------|
| R | A027750 |     |       |  1      |  1 2      |  1   3      |
| S | A027750 |     |       |  1      |  1 2      |  1   3      |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
| L | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
| I |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| N | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| K |         |  |  |  |\|  |  |\|\|  |  |\|\|\|  |  |\|\|\|\|  |
|   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
| P |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
| A |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| R |         |     |       |  3      |  3 1      |  3 1 1      |
| T |         |     |       |         |  2 2      |  2 2 1      |
| I |         |     |       |         |  4        |  4 1        |
| T |         |     |       |         |           |  3 2        |
| I |         |     |       |         |           |  5          |
| O |         |     |       |         |           |             |
| N |         |     |       |         |           |             |
| S |         |     |       |         |           |             |
|---|---------|-----|-------|---------|-----------|-------------|
.
The upper zone is a condensed version of the "divisors" zone.
The above table is the table of A340056 upside down.

Paolo Xausa, Table of n, a(n) for n = 1..11528 (rows 1..75 of the triangle, flattened)

Nonzero terms of A221649.

Cf. A000041, A002260, A027750, A066186, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A221529, A221530, A221531, A236104, A237593, A245092, A245095, A221650, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340011, A340031, A340032, A340056, A340057, A340061.

A340057row[n_]:=Flatten[Table[Divisors[m]PartitionsP[n-m],{m,n}]];Array[A340057row,10] (* Paolo Xausa, Sep 02 2023 *)

A346533 Irregular triangle read by rows in which row n lists the first n - 2 terms of A000203 together with the sum of A000203(n-1) and A000203(n), with a(1) = 1.

Original entry on oeis.org

1, 4, 1, 7, 1, 3, 11, 1, 3, 4, 13, 1, 3, 4, 7, 18, 1, 3, 4, 7, 6, 20, 1, 3, 4, 7, 6, 12, 23, 1, 3, 4, 7, 6, 12, 8, 28, 1, 3, 4, 7, 6, 12, 8, 15, 31, 1, 3, 4, 7, 6, 12, 8, 15, 13, 30, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 40, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 42, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 38

Offset: 1

Omar E. Pol, Jul 22 2021

T(n,k) is the total area (or number of cells) of the terraces that are in the k-th level that contains terraces starting from the top of the symmetric tower (a polycube) described in A221529.
The height of the tower equals A000041(n-1).
The terraces of the tower are the symmetric representation of sigma.
The terraces are in the levels that are the partition numbers A000041 starting from the base.
Note that for n >= 2 there are n - 1 terraces because the lower terrace of the tower is formed by two symmetric representations of sigma in the same level.

			Triangle begins:
  1;
  4;
  1, 7;
  1, 3, 11;
  1, 3,  4, 13;
  1, 3,  4,  7, 18;
  1, 3,  4,  7,  6, 20;
  1, 3,  4,  7,  6, 12, 23;
  1, 3,  4,  7,  6, 12,  8, 28;
  1, 3,  4,  7,  6, 12,  8, 15, 31;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 30;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 18, 40;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 18, 12, 42;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 18, 12, 28, 38;
  ...
For n = 7, sigma(7) = 1 + 7 = 8 and sigma(6) = 1 + 2 + 3 + 6 = 12, and 8 + 12 = 20, so the last term of row 7 is T(7,6) = 20. The other terms in row 7 are the first five terms of A000203, so the 7th row of the triangle is [1, 3, 4, 7, 6, 20].
For n = 7 we can see below the top view and the lateral view of the pyramid described in A245092 (with seven levels) and the top view and the lateral view of the tower described in A221529 (with 11 levels).
                                           _
                                          | |
                                          | |
                                          | |
        _                                 |_|_
       |_|_                               |   |
       |_ _|_                             |_ _|_
       |_ _|_|_                           |   | |
       |_ _ _| |_                         |_ _|_|_
       |_ _ _|_ _|_                       |_ _ _| |_
       |_ _ _ _| | |_                     |_ _ _|_ _|_ _
       |_ _ _ _|_|_ _|                    |_ _ _ _|_|_ _|
.
         Figure 1.                           Figure 2.
        Lateral view                       Lateral view
       of the pyramid.                     of the tower.
.
.       _ _ _ _ _ _ _                      _ _ _ _ _ _ _
       |_| | | | | | |                    |_| | | | |   |
       |_ _|_| | | | |                    |_ _|_| | |   |
       |_ _|  _|_| | |                    |_ _|  _|_|   |
       |_ _ _|    _|_|                    |_ _ _|    _ _|
       |_ _ _|  _|                        |_ _ _|  _|
       |_ _ _ _|                          |       |
       |_ _ _ _|                          |_ _ _ _|
.
          Figure 3.                          Figure 4.
          Top view                           Top view
       of the pyramid.                     of the tower.
.
Both polycubes have the same base which has an area equal to A024916(7) = 41 equaling the sum of the 7th row of triangle.
Note that in the top view of the tower the symmetric representation of sigma(6) and the symmetric representation of sigma(7) appear unified in the level 1 of the structure as shown above in the figure 4 (that is due the first two partition numbers A000041 are [1, 1]), so T(7,6) = sigma(7) + sigma(6) = 8 + 12 = 20.
.
Illustration of initial terms:
   Row 1    Row 2      Row 3      Row 4        Row 5          Row 6
.
    1        4         1 7        1 3 11       1 3 4 13       1 3 4 7 18
.   _        _ _       _ _ _      _ _ _ _      _ _ _ _ _      _ _ _ _ _ _
   |_|      |   |     |_|   |    |_| |   |    |_| | |   |    |_| | | |   |
            |_ _|     |    _|    |_ _|   |    |_ _|_|   |    |_ _|_| |   |
                      |_ _|      |      _|    |_ _|  _ _|    |_ _|  _|   |
                                 |_ _ _|      |     |        |_ _ _|    _|
                                              |_ _ _|        |        _|
                                                             |_ _ _ _|
.

Paolo Xausa, Table of n, a(n) for n = 1..11176 (rows 1..150 of the triangle, flattened)
Omar E. Pol, Illustration of the partitions of 10, prism and tower

Mirror of A340584.
The length of row n is A028310(n-1).
Row sums give A024916.
Leading diagonal gives A092403.
Other diagonals give A000203.
Companion of A346562.
Cf. A175254 (volume of the pyramid).
Cf. A066186 (volume of the tower).
Cf. A000041, A221529, A237270, A237593, A245092, A245093 (similar), A336811, A336812, A338156, A339106, A340035.

A346533row[n_]:=If[n==1,{1},Join[DivisorSigma[1,Range[n-2]],{Total[DivisorSigma[1,{n-1,n}]]}]];Array[A346533row,15] (* Paolo Xausa, Oct 23 2023 *)

A340527 Triangle read by rows: T(n,k) = A024916(n-k+1)*A000041(k-1), 1 <= k <= n.

Original entry on oeis.org

1, 4, 1, 8, 4, 2, 15, 8, 8, 3, 21, 15, 16, 12, 5, 33, 21, 30, 24, 20, 7, 41, 33, 42, 45, 40, 28, 11, 56, 41, 66, 63, 75, 56, 44, 15, 69, 56, 82, 99, 105, 105, 88, 60, 22, 87, 69, 112, 123, 165, 147, 165, 120, 88, 30, 99, 87, 138, 168, 205, 231, 231, 225, 176, 120, 42, 127, 99, 174

Offset: 1

Omar E. Pol, Jan 10 2021

Conjecture 1: T(n,k) is the sum of divisors of the terms that are in the k-th blocks of the first n rows of triangle A176206.
Conjecture 2: the sum of row n equals A182738(n), the sum of all parts of all partitions of all positive integers <= n.
Conjecture 3: T(n,k) is also the volume (or number of cubes) of the k-th block of a symmetric tower in which the terraces are the symmetric representation of sigma (n..1) starting from the base respectively (cf. A237270, A237593), hence the total area of the terraces is A024916(n), the same as the area of the base.
The levels of the terraces starting from the base are the first n terms of A000070, that is A000070(0)..A000070(n-1). Hence the differences between levels give the partition numbers A000041, that is A000041(0)..A000041(n-1).
This symmetric tower has the property that its volume (or total number of cubes) equals A182738(n), the sum of all parts of all partitions of all positive integers <= n.
For another symmetric tower of the same family and whose volume equals A066186(n) see A339106 and A221529.
The above three conjectures are connected due to the correspondence between divisors and partitions (cf. A336811).

			Triangle begins:
   1;
   4,   1;
   8,   4,   2;
  15,   8,   8,   3;
  21,  15,  16,  12,   5;
  33,  21,  30,  24,  20,   7;
  41,  33,  42,  45,  40,  28,  11;
  56,  41,  66,  63,  75,  56,  44,  15;
  69,  56,  82,  99, 105, 105,  88,  60,  22;
  87,  69, 112, 123, 165, 147, 165, 120,  88,  30;
  99,  87, 138, 168, 205, 231, 231, 225, 176, 120,  42;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A000041         T(6,k)
1      1  *  33   =   33
2      1  *  21   =   21
3      2  *  15   =   30
4      3  *   8   =   24
5      5  *   4   =   20
6      7  *   1   =    7
.          A024916
--------------------------
The sum of row 6 is 33 + 21 + 30 + 24 + 20 + 7 = 135, equaling A182738(6).

Columns 1 and 2 give A024916.
Column 3 gives A327329.
Leading diagonal gives A000041.
Row sums give A182738.
Cf. A000070, A066186, A176206, A221529, A221531, A237270, A237593, A336811, A336812, A338156, A339106, A340035,  A340424, A340425, A340426, A340524, A340526.

A340531 Irregular triangle read by rows T(n,k), (n >= 1, k >= 1), in which row n has length is A000070(n-1) and every column k is A024916, the sum of all divisors of all numbers <= n.

Original entry on oeis.org

1, 4, 1, 8, 4, 1, 1, 15, 8, 4, 4, 1, 1, 1, 21, 15, 8, 8, 4, 4, 4, 1, 1, 1, 1, 1, 33, 21, 15, 15, 8, 8, 8, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 41, 33, 21, 21, 15, 15, 15, 8, 8, 8, 8, 8, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 56, 41, 33, 33, 21, 21, 21, 15, 15, 15, 15, 15

Offset: 1

Omar E. Pol, Jan 10 2021

Consider a symmetric tower (a polycube) in which the terraces are the symmetric representation of sigma (n..1) respectively starting from the base (cf. A237270, A237593).
The levels of the terraces starting from the base are the first n terms of A000070, that is A000070(0)..A000070(n-1), hence the differences between two successive levels give the partition numbers A000041, that is A000041(0)..A000041(n-1).
T(n,k) is the volume (the number of cells) in the k-th level starting from the base.
This polycube has the property that the volume (the total number of cells) equals A182738(n), the sum of all parts of all partitions of all positive integers <= n.
A dissection of the symmetric tower is a three-dimensional spiral whose top view is described in A239660.
Other triangles related to the volume of this polycube are A340527 and A340579.
The symmetric tower is a member of the family of the stepped pyramid described in A245092.
For another symmetric tower of the same family and whose volume equals A066186(n) see A340423.
The sum of row n of triangle equals A182738(n). That property is due to the correspondence between divisors and parts. For more information see A336811.

			Triangle begins:
   1;
   4,  1;
   8,  4,  1,  1;
  15,  8,  4,  4, 1, 1, 1;
  21, 15,  8,  8, 4, 4, 4, 1, 1, 1, 1, 1;
  33, 21, 15, 15, 8, 8, 8, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1;
...
For n = 5 the length of row 5 is A000070(4) = 12.
The sum of row 5 is 21 + 15 + 8 + 8 + 4 + 4 + 4 + 1 + 1 + 1 + 1 + 1 = 69, equaling A182738(5).

Row sums give A182738.
Cf. A340527 (a regular version).
Members of the same family are: A176206, A337209, A339258, A340530.
Cf. A221529, A239660, A339278, A339304, A340423, A340529.
Cf. A000070, A024916, A237593, A336811, A338156.

a(m) = A024916(A176206(m)), assuming A176206 has offset 1.

T(n,k) = A024916(A176206(n,k)), assuming A176206 has offset 1.

A346741 Irregular triangle read by rows which is constructed in row n replacing the first A000070(n-1) terms of A336811 with their divisors.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 2, 1, 1, 5, 1, 3, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 2, 1, 1, 5, 1, 3, 1, 2, 1, 1, 1, 2, 3, 6, 1, 2, 4, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 5, 1, 2, 4, 1, 3, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Jul 31 2021

The terms in row n are also all parts of all partitions of n.
The terms of row n in nonincreasing order give the n-th row of A302246.
The terms of row n in nondecreasing order give the n-th row of A302247.
For further information about the correspondence divisor/part see A336811 and A338156.

			Triangle begins:
[1];
[1],[1, 2];
[1],[1, 2],[1, 3],[1];
[1],[1, 2],[1, 3],[1],[1, 2, 4],[1, 2],[1];
[1],[1, 2],[1, 3],[1],[1, 2, 4],[1, 2],[1],[1, 5],[1, 3],[1, 2],[1],[1];
...
Below the table shows the correspondence divisor/part.
|---|-----------------|-----|-------|---------|-----------|-------------|
| n |                 |  1  |   2   |    3    |     4     |      5      |
|---|-----------------|-----|-------|---------|-----------|-------------|
| P |                 |     |       |         |           |             |
| A |                 |     |       |         |           |             |
| R |                 |     |       |         |           |             |
| T |                 |     |       |         |           |  5          |
| I |                 |     |       |         |           |  3 2        |
| T |                 |     |       |         |  4        |  4 1        |
| I |                 |     |       |         |  2 2      |  2 2 1      |
| O |                 |     |       |  3      |  3 1      |  3 1 1      |
| N |                 |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| S |                 |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
----|-----------------|-----|-------|---------|-----------|-------------|
.
|---|-----------------|-----|-------|---------|-----------|-------------|
|   |         A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
| L |                 |  |  |  |/|  |  |/|/|  |  |/|/|/|  |  |/|/|/|/|  |
| I |         A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| N |                 |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| K |         A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
|   |                 |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
|   |         A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|---|-----------------|-----|-------|---------|-----------|-------------|
.
.   |-------|
.   |Section|
|---|-------|---------|-----|-------|---------|-----------|-------------|
|   |   1   | A000012 |  1  |  1    |  1      |  1        |  1          |
|   |-------|---------|-----|-------|---------|-----------|-------------|
|   |   2   | A000034 |     |  1 2  |  1 2    |  1 2      |  1 2        |
|   |-------|---------|-----|-------|---------|-----------|-------------|
| D |   3   | A010684 |     |       |  1   3  |  1   3    |  1   3      |
| I |       | A000012 |     |       |  1      |  1        |  1          |
| V |-------|---------|-----|-------|---------|-----------|-------------|
| I |   4   | A069705 |     |       |         |  1 2   4  |  1 2   4    |
| S |       | A000034 |     |       |         |  1 2      |  1 2        |
| O |       | A000012 |     |       |         |  1        |  1          |
| R |-------|---------|-----|-------|---------|-----------|-------------|
| S |   5   | A010686 |     |       |         |           |  1       5  |
|   |       | A010684 |     |       |         |           |  1   3      |
|   |       | A000034 |     |       |         |           |  1 2        |
|   |       | A000012 |     |       |         |           |  1          |
|   |       | A000012 |     |       |         |           |  1          |
|---|-------|---------|-----|-------|---------|-----------|-------------|
.
In the above table both the zone of partitions and the "Link" zone are the same zones as in the table of the example section of A338156, but here in the lower zone the divisors are ordered in accordance with the sections of the set of partitions of n.
The number of rows in the j-th section of the lower zone is equal to A000041(j-1).
The divisors of the j-th section are also the parts of the j-th section of the set of partitions of n.

Another version of A338156.
Row n has length A006128(n).
The sum of row n is A066186(n).
The product of row n is A007870(n).
Row n lists the first n rows of A336812.
The number of parts k in row n is A066633(n,k).
The sum of all parts k in row n is A138785(n,k).
The number of parts >= k in row n is A181187(n,k).
The sum of all parts >= k in row n is A206561(n,k).
The number of parts <= k in row n is A210947(n,k).
The sum of all parts <= k in row n is A210948(n,k).
Cf. A000012, A000034, A000041, A000070, A002260, A010684, A010686, A027750, A066633, A069705, A135010, A138785, A181187, A221529, A221649, A237593, A302246, A302247, A336811, A340011, A340031, A340032, A340035, A340056, A340057.

A346530 a(n) is the number of faces of the polycube called "tower" described in A221529 where n is the longest side of its base.

Original entry on oeis.org

6, 6, 11, 14, 20, 27, 31, 38, 42, 51, 59

Offset: 1

Omar E. Pol, Jul 22 2021

The tower is a geometric object associated to all partitions of n.

The height of the tower equals A000041(n-1).

			For n = 1 the tower is a cube, and a cube has 6 faces, so a(1) = 6.

Cf. A000203 (area of the terraces), A000041 (height of the terraces), A066186 (volume), A345023 (surface area), A346531 (number of edges), A346532 (number of vertices).
Cf. A325300 (analog for the pyramid described in A245092).
Cf. A221529, A236104, A237270, A237271, A237593, A336811, A338156, A340035, A340584.

a(n) = A346531(n) - A346532(n) + 2 (Euler's formula).

A346531 a(n) is the number of edges of the polycube called "tower" described in A221529 where n is the longest side of its base.

Original entry on oeis.org

12, 12, 27, 36, 51, 72, 84, 105, 117, 144, 165

Offset: 1

Omar E. Pol, Jul 22 2021

The tower is a geometric object associated to all partitions of n.

The height of the tower equals A000041(n-1).

			For n = 1 the tower is a cube, and a cube has 12 edges, so a(1) = 12.

Cf. A000203 (area of the terraces), A000041 (height of the terraces), A066186 (volume), A345023 (surface area), A346530 (number of faces), A346532 (number of vertices).
Cf. A325301 (analog for the pyramid described in A245092).
Cf. A221529, A236104, A237270, A237271, A237593, A336811, A338156, A340035, A340584.

a(n) = A346530(n) + A346532(n) - 2 (Euler's formula).

A346532 a(n) is the number of vertices of the polycube called "tower" described in A221529 where n is the longest side of its base.

Original entry on oeis.org

8, 8, 18, 24, 33, 47, 55, 69, 77, 95, 108

Offset: 1

Omar E. Pol, Jul 22 2021

The height of the tower equals A000041(n-1).

			For n = 1 the tower is a cube, and a cube has 8 vertices, so a(1) = 8.

Cf. A000203 (area of the terraces), A000041 (height of the terraces), A066186 (volume), A345023 (surface area), A346530 (number of faces), A346531 (number of edges).
Cf. A325302 (analog for the pyramid described in A245092).
Cf. A221529, A236104, A237270, A237271, A237593, A336811, A338156, A340035, A340584.

a(n) = A346531(n) - A346530(n) + 2 (Euler's formula).

A346562 Irregular triangle read by rows in which row n lists the first n - 2 terms of A000005 together with the sum of A000005(n-1) and A000005(n), with a(1) = 1.

Original entry on oeis.org

1, 3, 1, 4, 1, 2, 5, 1, 2, 2, 5, 1, 2, 2, 3, 6, 1, 2, 2, 3, 2, 6, 1, 2, 2, 3, 2, 4, 6, 1, 2, 2, 3, 2, 4, 2, 7, 1, 2, 2, 3, 2, 4, 2, 4, 7, 1, 2, 2, 3, 2, 4, 2, 4, 3, 6, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 8, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 8, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 6

Offset: 1

Omar E. Pol, Jul 23 2021

T(n,k) is the total number of divisors related to the terraces that are in the k-th level that contains terraces starting from the base of the symmetric tower described in A221529.

			Triangle begins:
1;
3;
1, 4;
1, 2, 5;
1, 2, 2, 5;
1, 2, 2, 3, 6;
1, 2, 2, 3, 2, 6;
1, 2, 2, 3, 2, 4, 6;
1, 2, 2, 3, 2, 4, 2, 7;
1, 2, 2, 3, 2, 4, 2, 4, 7;
1, 2, 2, 3, 2, 4, 2, 4, 3, 6;
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 8;
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 8;
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 6;
...

The length of row n is A028310(n-1).
Row sums give A006218, n >= 1.
Leading diagonal gives A092405.
Other diagonals give A000005.
Column 1 gives the absolute values of A260196.
Companion of A346533.
Cf. A221529, A237270, A237593, A336811, A338156, A340035.

cp's OEIS Frontend

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A340057 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the block m consists of the divisors of m multiplied by A000041(n-m), with 1 <= m <= n.

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A346533 Irregular triangle read by rows in which row n lists the first n - 2 terms of A000203 together with the sum of A000203(n-1) and A000203(n), with a(1) = 1.

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A340527 Triangle read by rows: T(n,k) = A024916(n-k+1)*A000041(k-1), 1 <= k <= n.

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A340531 Irregular triangle read by rows T(n,k), (n >= 1, k >= 1), in which row n has length is A000070(n-1) and every column k is A024916, the sum of all divisors of all numbers <= n.

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A346741 Irregular triangle read by rows which is constructed in row n replacing the first A000070(n-1) terms of A336811 with their divisors.

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A346530 a(n) is the number of faces of the polycube called "tower" described in A221529 where n is the longest side of its base.

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A346531 a(n) is the number of edges of the polycube called "tower" described in A221529 where n is the longest side of its base.

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A346532 a(n) is the number of vertices of the polycube called "tower" described in A221529 where n is the longest side of its base.

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A346562 Irregular triangle read by rows in which row n lists the first n - 2 terms of A000005 together with the sum of A000005(n-1) and A000005(n), with a(1) = 1.

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