A340061 -id:A340061 - 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 *)

A340492 a(n) = A000041(n)*A000070(n-1), n >= 1.

Original entry on oeis.org

1, 4, 12, 35, 84, 209, 450, 990, 2010, 4074, 7784, 15015, 27472, 50355, 89408, 158004, 271755, 466620, 782530, 1308549, 2149488, 3513012, 5657540, 9076725, 14367804, 22645056, 35313320, 54810756, 84269900, 129032100, 195879618, 296147379, 444466260, 664284530, 986341059, 1458941412

Offset: 1

Omar E. Pol, Jan 10 2021

nonn

Conjecture: a(n) is the size of a table of correspondence between the divisors of certain numbers and the partitions of n. The table contains A006128(n) positive integers and the rest are zeros (or empty cells). These positive integers can be interpreted in two ways as follows:
1) They are all divisors of all terms of the first n rows of triangle A336811. In other words: they are all divisors of the first A000070(n-1) terms of A336811, hence they are all divisors of all terms in the n-th row of A176206.
2) They are all parts of all partitions of n.
The mentioned divisors are in the columns of the table.
The mentioned partitions are in the rows of the table.
For more information about the main conjecture see A336811.

			Illustration of initial terms:
       A000070:  1 2   4     7        12            19                    30
A000041         _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
     1         |_| |   |     |         |             |                     |
     2         |_ _|   |     |         |             |                     |
     3         |_ _ _ _|     |         |             |                     |
               |             |         |             |                     |
     5         |_ _ _ _ _ _ _|         |             |                     |
               |                       |             |                     |
     7         |_ _ _ _ _ _ _ _ _ _ _ _|             |                     |
               |                                     |                     |
               |                                     |                     |
               |                                     |                     |
    11         |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|                     |
               |                                                           |
               |                                                           |
               |                                                           |
    15         |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
...
a(n) is the area (or the number of cells) in the n-th rectangle of the diagram.
For n = 3 the third rectangle of the diagram contains 3*4 = 12 cells, so a(3) = 12
For n = 7 the seventh rectangle of the diagram contains 15*30 = 450 cells, so a(7) = 450.

Partial sums of A340493.

Cf. A000041, A000070, A006128, A176206, A336811, A340061.

a[n_] := PartitionsP[n]*Count[Flatten[IntegerPartitions[n]], 1]; Table[a[n], {n, 1, 36}] (* Robert P. P. McKone, Jan 28 2021 *)

a(n) = numbpart(n)*sum(k=0, n-1, numbpart(k)); \\ Michel Marcus, Jan 28 2021

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

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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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A340492 a(n) = A000041(n)*A000070(n-1), n >= 1.

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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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