A339278 -id:A339278 - cp's OEIS frontend

A340056 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the divisors of j multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.

1, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 4, 1, 3, 2, 4, 3, 1, 5, 1, 2, 4, 2, 6, 3, 6, 5, 1, 2, 3, 6, 1, 5, 2, 4, 8, 3, 9, 5, 10, 7, 1, 7, 1, 2, 3, 6, 2, 10, 3, 6, 12, 5, 15, 7, 14, 11, 1, 2, 4, 8, 1, 7, 2, 4, 6, 12, 3, 15, 5, 10, 20, 7, 21, 11, 22, 15, 1, 3, 9, 1, 2, 4, 8, 2, 14, 3, 6, 9, 18, 5

Offset: 1

Omar E. Pol, Dec 27 2020

This triangle is a condensed version of the more irregular triangle A338156 which is the main sequence with further information about the correspondence divisor/part.

			Triangle begins:
  [1];
  [1, 2],    [1];
  [1, 3],    [1, 2],    [2];
  [1, 2, 4], [1, 3],    [2, 4], [3];
  [1, 5],    [1, 2, 4], [2, 6], [3, 6], [5];
  [...
The row sums of triangle give A066186.
Written as an irregular tetrahedron the first five slices are:
  1;
  -----
  1, 2,
  1;
  -----
  1, 3,
  1, 2,
  2;
  --------
  1, 2, 4,
  1, 3,
  2, 4,
  3;
  --------
  1, 5,
  1, 2, 4,
  2, 6,
  3, 6,
  5;
  --------
The row sums of tetrahedron give A339106.
The slices of the tetrahedron appear in the following table formed by four zones shows the correspondence between divisor and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| 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  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |
|   |---------|-----|-------|---------|-----------|-------------|
| D | A027750 |     |       |  1      |  1 2      |  1   3      |
| I | A027750 |     |       |  1      |  1 2      |  1   3      |
| V |---------|-----|-------|---------|-----------|-------------|
| I | A027750 |     |       |         |  1        |  1 2        |
| S | A027750 |     |       |         |  1        |  1 2        |
| O | A027750 |     |       |         |  1        |  1 2        |
| R |---------|-----|-------|---------|-----------|-------------|
| S | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |
| C | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |
| O |    -    |     |       |  2      |  2 4      |  2   6      |
| N |    -    |     |       |         |  3        |  3 6        |
| D |    -    |     |       |         |           |  5          |
|---|---------|-----|-------|---------|-----------|-------------|
.
The lower zone is a condensed version of the "divisors" zone.

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

Nonzero terms of A340011.
Row sums give A066186.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A245095, A221649, A221650, A237593, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340061.

A340056row[n_]:=Flatten[Table[Divisors[n-m]PartitionsP[m],{m,0,n-1}]];Array[A340056row,10] (* Paolo Xausa, Sep 01 2023 *)

A340423 Irregular triangle read by rows T(n,k) in which row n has length A000041(n-1) and every column k is A024916, n >= 1, k >= 1.

Original entry on oeis.org

1, 4, 8, 1, 15, 4, 1, 21, 8, 4, 1, 1, 33, 15, 8, 4, 4, 1, 1, 41, 21, 15, 8, 8, 4, 4, 1, 1, 1, 1, 56, 33, 21, 15, 15, 8, 8, 4, 4, 4, 4, 1, 1, 1, 1, 69, 41, 33, 21, 21, 15, 15, 8, 8, 8, 8, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 87, 56, 41, 33, 33, 21, 21, 15, 15, 15, 15, 8, 8, 8, 8

Offset: 1

Omar E. Pol, Jan 07 2021

T(n,k) is the number of cubic cells (or cubes) in the k-th level starting from the base of the tower described in A221529 whose largest side of the base is equal to n (see example). - Omar E. Pol, Jan 08 2022

			Triangle begins:
   1;
   4;
   8,  1;
  15,  4,  1;
  21,  8,  4,  1,  1;
  33, 15,  8,  4,  4,  1,  1;
  41, 21, 15,  8,  8,  4,  4, 1, 1, 1, 1;
  56, 33, 21, 15, 15,  8,  8, 4, 4, 4, 4, 1, 1, 1, 1;
  69, 41, 33, 21, 21, 15, 15, 8, 8, 8, 8, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1;
...
For n = 9 the length of row 9 is A000041(9-1) = 22.
From _Omar E. Pol_, Jan 08 2022: (Start)
For n = 9 the lateral view and top view of the tower described in A221529 look like as shown below:
                        _
    22        1        | |
    21        1        | |
    20        1        | |
    19        1        | |
    18        1        | |
    17        1        | |
    16        1        |_|_
    15        4        |   |
    14        4        |   |
    13        4        |   |
    12        4        |_ _|_
    11        8        |   | |
    10        8        |   | |
     9        8        |   | |
     8        8        |_ _|_|_
     7       15        |     | |
     6       15        |_ _ _| |_
     5       21        |     |   |
     4       21        |_ _ _|_ _|_
     3       33        |_ _ _ _| | |_
     2       41        |_ _ _ _|_|_ _|_ _
     1       69        |_ _ _ _ _|_ _|_ _|
.
   Level   Row 9         Lateral view
     k     T(9,k)        of the tower
.
                        _ _ _ _ _ _ _ _ _
                       |_| | | | | | |   |
                       |_ _|_| | | | |   |
                       |_ _|  _|_| | |   |
                       |_ _ _|    _|_|   |
                       |_ _ _|  _|    _ _|
                       |_ _ _ _|     |
                       |_ _ _ _|  _ _|
                       |         |
                       |_ _ _ _ _|
.
                           Top view
                         of the tower
.
For n = 9 and k = 1 there are 69 cubic cells in the level 1 starting from the base of the tower, so T(9,1) = 69.
For n = 9 and k = 22 there is only one cubic cell in the level 22 (the top) of the tower, so T(9,22) = 1.
The volume of the tower (also the total number of cubic cells) represents the 9th term of the convolution of A000203 and A000041 hence it's equal to A066186(9) = 270, equaling the sum of the 9th row of triangle. (End)

Row sums give A066186.
Row lengths give A000041.
The length of the m-th block in row n is A187219(m), m >= 1.
Cf. A350637 (analog for the stepped pyramid described in A245092).
Cf. A000203, A024916, A196020, A221529, A236104, A235791, A237270, A237271, A237593, A339278, A262626, A336811, A338156, A340035, A341149, A346533, A350333.

f(n) = numbpart(n-1);
T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (n)); my(s=0); while (k <= f(n-1), s++; n--; ); 1+s; } \\ A336811
g(n) = sum(k=1, n, n\k*k); \\ A024916
row(n) = vector(f(n), k, g(T(n,k))); \\ Michel Marcus, Jan 22 2022

T(n,k) = A024916(A336811(n,k)).

T(n,k) = Sum_{j=1..n} A339278(j,k). - Omar E. Pol, Jan 08 2022

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 *)

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.

A340583 Triangle read by rows: T(n,k) = A002865(n-k)*A000203(k), 1 <= k <= n.

Original entry on oeis.org

1, 0, 3, 1, 0, 4, 1, 3, 0, 7, 2, 3, 4, 0, 6, 2, 6, 4, 7, 0, 12, 4, 6, 8, 7, 6, 0, 8, 4, 12, 8, 14, 6, 12, 0, 15, 7, 12, 16, 14, 12, 12, 8, 0, 13, 8, 21, 16, 28, 12, 24, 8, 15, 0, 18, 12, 24, 28, 28, 24, 24, 16, 15, 13, 0, 12, 14, 36, 32, 49, 24, 48, 16, 30, 13, 18, 0, 28

Offset: 1

Omar E. Pol, Jan 15 2021

T(n,k) is the total number of cubic cells added at n-th stage to the right prisms whose bases are the parts of the symmetric representation of sigma(k) in the polycube described in A221529.

Partial sums of column k gives the column k of A221529.

			Triangle begins:
   1;
   0,  3;
   1,  0,  4;
   1,  3,  0,  7;
   2,  3,  4,  0,  6;
   2,  6,  4,  7,  0, 12;
   4,  6,  8,  7,  6,  0,  8;
   4, 12,  8, 14,  6, 12,  0, 15;
   7, 12, 16, 14, 12, 12,  8,  0, 13;
   8, 21, 16, 28, 12, 24,  8, 15,  0, 18;
  12, 24, 28, 28, 24, 24, 16, 15, 13,  0, 12;
  14, 36, 32, 49, 24, 48, 16, 30, 13, 18,  0, 28;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A000203         T(6,k)
--------------------------
1      1   *   2  =    2
2      3   *   2   =   6
3      4   *   1   =   4
4      7   *   1   =   7
5      6   *   0   =   0
6     12   *   1   =  12
.           A002865
--------------------------
The sum of row 6 is 2 + 6 + 4 + 7 + 0 + 12 = 31, equaling A138879(6).

Row sums give A138879.
Column 1 gives A002865.
Diagonals 1, 3 and 4 give A000203.
Diagonal 2 gives A000004.
Diagonals 5 and 6 give A074400.
Diagonals 7 and 8 give A239050.
Diagonal 9 gives A319527.
Diagonal 10 gives A319528.
Cf. A221529 (partial column sums).
Cf. A340426 (mirror).
Cf. A135010, A221530, A245095, A245099, A339278.

A340583[n_, k_] := (PartitionsP[n - k] - PartitionsP[(n - k) - 1])*
   DivisorSigma[1, k];
Table[A340583[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Robert P. P. McKone, Jan 25 2021 *)

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.

A350357 Irregular triangle read by rows in which row n lists all elements of the arrangement of the correspondence divisor/part related to the last section of the set of partitions of n in the following order: row n lists the n-th row of A138121 followed by the n-th row of A336812.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 26 2021

			Triangle begins:
[1], [1];
[2, 1], [1, 2];
[3, 1, 1], [1, 3, 1];
[4, 2, 2, 1, 1, 1], [1, 2, 4, 1, 2, 1];
[5, 3, 2, 1, 1, 1, 1, 1], [1, 5, 1, 3, 1, 2, 1, 1];
...
Illustration of the first six rows of triangle in an infinite table:
|---|---------|-----|-------|---------|-----------|-------------|---------------|
| n |         |  1  |   2   |    3    |     4     |      5      |       6       |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
|   |         |     |       |         |           |             |  6            |
|   |         |     |       |         |           |             |  3 3          |
|   |         |     |       |         |           |             |  4 2          |
| P |         |     |       |         |           |             |  2 2 2        |
| A |         |     |       |         |           |  5          |    1          |
| R |         |     |       |         |           |  3 2        |      1        |
| T |         |     |       |         |  4        |    1        |      1        |
| S |         |     |       |         |  2 2      |      1      |        1      |
|   |         |     |       |  3      |    1      |      1      |        1      |
|   |         |     |  2    |    1    |      1    |        1    |          1    |
|   |         |  1  |    1  |      1  |        1  |          1  |            1  |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
| D | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |  1 2 3     6  |
| I | A027750 |     |       |  1      |  1 2      |  1   3      |  1 2   4      |
| V | A027750 |     |       |         |  1        |  1 2        |  1   3        |
| I | A027750 |     |       |         |           |  1          |  1 2          |
| S | A027750 |     |       |         |           |  1          |  1 2          |
| O | A027750 |     |       |         |           |             |  1            |
| R | A027750 |     |       |         |           |             |  1            |
| S |         |     |       |         |           |             |               |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
.
For n = 6 in the upper zone of the above table we can see the parts of the last section of the set of partitions of 6 in reverse-colexicographic order in accordance with the 6th row of A138121.
In the lower zone of the table we can see the terms from the 6th row of A336812, these are the divisors of the numbers from the 6th row of A336811.
Note that in the lower zone of the table every row gives A027750.
The remarkable fact is that the elements in the lower zone of the arrangement are the same as the elements in the upper zone but in other order.
For an explanation of the connection of the elements of the upper zone with the elements of the lower zone, that is the correspondence divisor/part, see A336812 and A338156.
The growth of the upper zone of the table is in accordance with the growth of the modular prism described in A221529.
The growth of the lower zone of the table is in accordance with the growth of the tower described also in A221529.
The number of cubic cells added at n-th stage in each polycube is equal to A138879(10) = 150, hence the total number of cubic cells added at n-th stage is equal to 2*A138879(10) = 300, equaling the sum of the 10th row of this triangle.

Companion of A350333.
Row sums give 2*A138879.
Row lengths give 2*A138137.
Cf. A002865, A135010, A138121, A138879, A221529, A237593, A336812, A339278.

A341049 Irregular triangle read by rows T(n,k) in which row n lists the terms of n-th row of A336811 in nondecreasing order.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 4, 1, 1, 2, 3, 5, 1, 1, 2, 2, 3, 4, 6, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 6, 8, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 6, 7, 9, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 7, 8, 10

Offset: 1

Omar E. Pol, Feb 04 2021

All divisors of all terms of n-th row are also all parts of the last section of the set of partitions of n.
All divisors of all terms of the first n rows are also all parts of all partitions of n. In other words: all divisors of the first A000070(n-1) terms of the sequence are also all parts of all partitions of n.
For further information about the correspondence divisor/part see A338156 and A336812.

			Triangle begins:
1;
2;
1, 3;
1, 2, 4;
1, 1, 2, 3, 5;
1, 1, 2, 2, 3, 4, 6;
1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 7;
1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 6, 8;
1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 6, 7, 9;
...

Paolo Xausa, Table of n, a(n) for n = 1..11732 (rows 1..27 of triangle, flattened)

Mirror of A336811.
Row n has length A000041(n-1).
Row sums give A000070.
Right border gives A000027.
Cf. A000070, A000041, A002865, A027750, A028310, A133735, A135010, A138121, A138137, A176206, A182703, A187219, A207378, A237593, A336812, A338156, A339278, A340061.

A341049[rowmax_]:=Table[Flatten[Table[ConstantArray[n-m,PartitionsP[m]-PartitionsP[m-1]],{m,n-1,0,-1}]],{n,rowmax}];
A341049[10] (* Generates 10 rows *) (* Paolo Xausa, Feb 17 2023 *)

A341049(rowmax)=vector(rowmax,n,concat(vector(n,m,vector(numbpart(n-m)-numbpart(n-m-1),i,m))));
A341049(10) \\ Generates 10 rows - Paolo Xausa, Feb 17 2023

cp's OEIS Frontend

A340056 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the divisors of j multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.

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A340423 Irregular triangle read by rows T(n,k) in which row n has length A000041(n-1) and every column k is A024916, n >= 1, k >= 1.

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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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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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A340583 Triangle read by rows: T(n,k) = A002865(n-k)*A000203(k), 1 <= k <= 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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A350357 Irregular triangle read by rows in which row n lists all elements of the arrangement of the correspondence divisor/part related to the last section of the set of partitions of n in the following order: row n lists the n-th row of A138121 followed by the n-th row of A336812.

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A341049 Irregular triangle read by rows T(n,k) in which row n lists the terms of n-th row of A336811 in nondecreasing order.

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