A337209 -id:A337209 - cp's OEIS frontend

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.

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 *)

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.

A340529 Irregular triangle read by rows T(n,k), (n >= 1, k >= 1), in which row n has length A000041(n-1) and every column k is A006218.

Original entry on oeis.org

1, 3, 5, 1, 8, 3, 1, 10, 5, 3, 1, 1, 14, 8, 5, 3, 3, 1, 1, 16, 10, 8, 5, 5, 3, 3, 1, 1, 1, 1, 20, 14, 10, 8, 8, 5, 5, 3, 3, 3, 3, 1, 1, 1, 1, 23, 16, 14, 10, 10, 8, 8, 5, 5, 5, 5, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 27, 20, 16, 14, 14, 10, 10, 8, 8, 8, 8, 5, 5, 5, 5, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Jan 10 2021

			Triangle begins:
   1;
   3;
   5,  1;
   8,  3,  1;
  10,  5,  3,  1,  1;
  14,  8,  5,  3,  3, 1, 1;
  16, 10,  8,  5,  5, 3, 3, 1, 1, 1, 1;
  20, 14, 10,  8,  8, 5, 5, 3, 3, 3, 3, 1, 1, 1, 1;
  23, 16, 14, 10, 10, 8, 8, 5, 5, 5, 5, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1;
...
For n = 6, the length of row 6 is A000041(5) = 7.
The sum of row 6 is 14 + 8 + 5 + 3 + 3 + 1 + 1 = 35, equaling A006128(6).

Row sums give A006128.
Cf. A340525 (a regular version).
Members of the same family are: A336811, A339278, A339304, A340423.
Cf. A337209, A339258, A340530, A340531.
Cf. A000041, A006218.

a(m) = A006218(A336811(m)).

T(n,k) = A006218(A336811(n,k)).

A340530 Irregular triangle read by rows T(n,k) in which row n has length is A000070(n-1) and every column k is A006218, (n >= 1, k >= 1).

Original entry on oeis.org

1, 3, 1, 5, 3, 1, 1, 8, 5, 3, 3, 1, 1, 1, 10, 8, 5, 5, 3, 3, 3, 1, 1, 1, 1, 1, 14, 10, 8, 8, 5, 5, 5, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 16, 14, 10, 10, 8, 8, 8, 5, 5, 5, 5, 5, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 20, 16, 14, 14, 10, 10, 10, 8, 8, 8, 8, 8, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Omar E. Pol, Jan 10 2021

The sum of row n equals A284870(n), the total number of parts in all partitions of all positive integers <= n. It is conjectured that this property is due to the correspondence between divisors and partitions. For more information see A336811.

			Triangle begins:
   1;
   3,  1;
   5,  3,  1,  1;
   8,  5,  3,  3, 1, 1, 1;
  10,  8,  5,  5, 3, 3, 3, 1, 1, 1, 1, 1;
  14, 10,  8,  8, 5, 5, 5, 3, 3, 3, 3, 3, 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 10 + 8 + 5 + 5 + 3 + 3 + 3 + 1 + 1 + 1 + 1 + 1 = 42, equaling A284870(5).

Row sums give A284870.
Cf. A340526 (a regular version).
Members of the same family are: A176206, A337209, A339258, A340531.
Cf. A339278, A339304, A340423, A340529.
Cf. A000070, A006128, A336811.

a(m) = A006218(A176206(m)), assuming A176206 has offset 1.

T(n,k) = A006218(A176206(n,k)), assuming A176206 has offset 1.

A341149 Irregular triangle read by rows T(n,k) in which row n lists n blocks where the m-th block consists of A000203(m) copies of A000041(n-m), with 1 <= m <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 5, 5, 5, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 7, 7, 7, 5, 5, 5, 5, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Omar E. Pol, Feb 06 2021

In the n-th row of the triangle the values of the m-th block are the number of cubes that are exactly below every cell of the symmetric representation of sigma(m) in the tower described in A221529 (see figure 5 in the example here).

			Triangle begins:
  1;
  1,1,1,1;
  2,1,1,1,1,1,1,1;
  3,2,2,2,1,1,1,1,1,1,1,1,1,1,1;
  5,3,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1;
  7,5,5,5,3,3,3,3,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
  ...
For n = 6 we have that:
                                 Row 6                    Row 6 of
m    A000203(m)  A000041(n-m)   block(m)                  A221529
1        1           7          [7]                           7
2        3           5          [5,5,5]                      15
3        4           3          [3,3,3,3]                    12
4        7           2          [2,2,2,2,2,2,2]              14
5        6           1          [1,1,1,1,1,1]                 6
6       12           1          [1,1,1,1,1,1,1,1,1,1,1,1]    12
.
so the 6th row of triangle is [7,5,5,5,3,3,3,3,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] and the row sums equals A066186(6) = 66.
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). For further information about these two associated objects see A221529.
       _ _ _ _ _ _
  11  |_ _ _      |              6
      |_ _ _|_    |        3     3
      |_ _    |   |          4   2
      |_ _|_ _|_  |      2   2   2      _
   7  |_ _ _    | |            5 1     | |
      |_ _ _|_  | |        3   2 1     |_|_
   5  |_ _    | | |          4 1 1     |   |
      |_ _|_  | | |      2   2 1 1     |_ _|_
   3  |_ _  | | | |        3 1 1 1     |_ _|_|_
   2  |_  | | | | |      2 1 1 1 1     |_ _ _|_|_ _
   1  |_|_|_|_|_|_|    1 1 1 1 1 1     |_ _ _ _|_|_|
.
        Figure 1.        Figure 2.       Figure 3.
       Front view       Partitions     Lateral view
      of the prism         of 6.       of the tower.
      of partitions.
.
                                                                      Row 6 of
                                        _ _ _ _ _ _                    A341148
                                    1  |_| | | |   |    7 5 3 2 1 1       19
                                    2  |_ _|_| |   |    5 5 3 2 1 1       17
                                    3  |_ _|  _|   |    3 3 2 2 1 1       12
                                    4  |_ _ _|    _|    2 2 2 1 1 1        9
                                    5  |        _|      1 1 1 1 1          5
                                    6  |_ _ _ _|        1 1 1 1            4
.
                                         Figure 4.       Figure 5.
                                         Top view         Heights
                                       of the tower.      in the
                                                         top view.
.
Figure 5 shows the heights of the terraces of the tower, or in other words the number of cubes in the column exactly below every cell of the top view. For example: in the 6th row of triangle the first block is [7] because there are seven cubes exactly below the symmetric representation of sigma(1) = 1. The second block is [5, 5, 5] because there are five cubes exactly below every cell of the symmetric representation of sigma(2) = 3. The third block is [3, 3, 3, 3] because there are three cubes exactly below every cell of the symmetric representation of sigma(3) = 4, and so on.
Note that the terraces that are the symmetric representation of sigma(5) and the terraces that are the symmetric representation of sigma(6) both are unified in level 1 of the structure. That is because the first two partition numbers A000041 are [1, 1].

Paolo Xausa, Table of n, a(n) for n = 1..12451 (rows 1..35 of triangle, flattened)

Every column gives A000041.
Row lengths give A024916.
Row sums give the nonzero terms of A066186.
Cf. A000203, A221529, A236104, A237270, A237271, A237593, A337209, A339106, A340584, A341148, A345023.

A341149row[n_]:=Flatten[Array[ConstantArray[PartitionsP[n-#],DivisorSigma[1,#]]&,n]];
nrows=7;Array[A341149row,nrows] (* Paolo Xausa, Jun 20 2022 *)

cp's OEIS Frontend

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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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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A340529 Irregular triangle read by rows T(n,k), (n >= 1, k >= 1), in which row n has length A000041(n-1) and every column k is A006218.

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A340530 Irregular triangle read by rows T(n,k) in which row n has length is A000070(n-1) and every column k is A006218, (n >= 1, k >= 1).

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A341149 Irregular triangle read by rows T(n,k) in which row n lists n blocks where the m-th block consists of A000203(m) copies of A000041(n-m), with 1 <= m <= n.

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