cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A338156 Irregular triangle read by rows in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the divisors of (n - m + 1), with 1 <= m <= n.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Oct 14 2020

Keywords

Comments

In other words: in row n replace every term of n-th row of A176206 with its divisors.
The terms in row n are also all parts of all partitions of n.
As in A336812 here we introduce a new type of table which shows the correspondence between divisors and partitions. More precisely the table shows the correspondence between all divisors of all terms of the n-th row of A176206 and all parts of all partitions of n, with n >= 1. Both the mentionded divisors and the mentioned parts are the same numbers (see Example section). That is because all divisors of the first A000070(n-1) terms of A336811 are also all parts of all partitions of n.
For an equivalent table for all parts of the last section of the set of partitions of n see the subsequence A336812. The section is the smallest substructure of the set of partitions in which appears the correspondence divisor/part.
From Omar E. Pol, Aug 01 2021: (Start)
The terms of row n appears in the triangle A346741 ordered in accordance with the successive sections of the set of 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 the connection with the tower described in A221529 see also A340035. (End)

Examples

			Triangle begins:
  [1];
  [1,2],   [1];
  [1,3],   [1,2],   [1],   [1];
  [1,2,4], [1,3],   [1,2], [1,2], [1],   [1],   [1];
  [1,5],   [1,2,4], [1,3], [1,3], [1,2], [1,2], [1,2], [1], [1], [1], [1], [1];
  ...
For n = 5 the 5th row of A176206 is [5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1] so replacing every term with its divisors we have the 5th row of this triangle.
Also, if the sequence is written as an irregular tetrahedron so the first six slices are:
  [1],
  -------
  [1, 2],
  [1],
  -------
  [1, 3],
  [1, 2],
  [1],
  [1];
  ----------
  [1, 2, 4],
  [1, 3],
  [1, 2],
  [1, 2],
  [1],
  [1],
  [1];
  ----------
  [1, 5],
  [1, 2, 4],
  [1, 3],
  [1, 3],
  [1, 2],
  [1, 2],
  [1, 2],
  [1],
  [1],
  [1],
  [1],
  [1];
.
The above slices appear in the lower zone of the following table which shows the correspondence between the mentioned divisors and all parts of all partitions of the positive integers.
The table is infinite. It is formed by three zones as follows:
The upper zone shows the partitions of every positive integer in colexicographic order (cf. A026792, A211992).
The lower zone shows the same numbers but arranged as divisors in accordance with the slices of the tetrahedron mentioned above.
Finally the middle zone shows the connection between the upper zone and the lower zone.
For every positive integer the numbers in the upper zone are the same numbers as in the lower zone.
.
|---|---------|-----|-------|---------|------------|---------------|
| 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 | A066633 |  1  |  2 1  |  4 1 1  |  7  3 1 1  | 12  4  2 1 1  |
| I |         |  *  |  * *  |  * * *  |  *  * * *  |  *  *  * * *  |
| N | A002260 |  1  |  1 2  |  1 2 3  |  1  2 3 4  |  1  2  3 4 5  |
| K |         |  =  |  = =  |  = = =  |  =  = = =  |  =  =  = = =  |
|   | A138785 |  1  |  2 2  |  4 2 3  |  7  6 3 4  | 12  8  6 4 5  |
|   |         |  |  |  |\|  |  |\|\|  |  |\ |\|\|  |  |\ |\ |\|\|  |
|   | A206561 |  1  |  4 2  |  9 5 3  | 20 13 7 4  | 35 23 15 9 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            |
|---|---------|-----|-------|---------|------------|---------------|
.
Note that every row in the lower zone lists A027750.
Also the lower zone for every positive integer can be constructed using the first n terms of the partition numbers. For example: for n = 5 we consider the first 5 terms of A000041 (that is [1, 1, 2, 3, 5]) then the 5th slice is formed by a block with the divisors of 5, one block with the divisors of 4, two blocks with the divisors of 3, three blocks with the divisors of 2, and five blocks with the divisors of 1.
Note that the lower zone is also in accordance with the tower (a polycube) described in A221529 in which its terraces are the symmetric representation of sigma starting from the top (cf. A237593) and the heights of the mentioned terraces are the partition numbers A000041 starting from the base.
The tower has the same volume (also the same number of cubes) equal to A066186(n) as a prism of partitions of size 1*n*A000041(n).
The above table shows the correspondence between the prism of partitions and its associated tower since the number of parts in all partitions of n is equal to A006128(n) equaling the number of divisors in the n-th slice of the lower table and equaling the same the number of terms in the n-th row of triangle. Also the sum of all parts of all partitions of n is equal to A066186(n) equaling the sum of all divisors in the n-th slice of the lower table and equaling the sum of the n-th row of triangle.

Links

Crossrefs

Nonzero terms of A340031.
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 (a subsequence).
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. A000070, A000041, A002260, A026792, A027750, A058399, A127093, A135010, A138121, A176206, A182703, A206437, A207031, A207383, A211992, A221529, A221530, A221531, A245095, A221649, A221650, A237593, A302246, A302247, A336811, A337209, A339106, A339258, A339278, A339304, A340035, A340061, A346741.

Programs

  • Mathematica
    A338156[rowmax_]:=Table[Flatten[Table[ConstantArray[Divisors[n-m],PartitionsP[m]],{m,0,n-1}]],{n,rowmax}];
A338156[10] (* Generates 10 rows *) (* Paolo Xausa, Jan 12 2023 *)
  • PARI
    A338156(rowmax)=vector(rowmax,n,concat(vector(n,m,concat(vector(numbpart(m-1),i,divisors(n-m+1))))));
A338156(10) \\ Generates 10 rows - Paolo Xausa, Feb 17 2023

A302246 Irregular triangle read by rows in which row n lists all parts of all partitions of n, in nonincreasing order.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Apr 05 2018

Keywords

Comments

Also due to the correspondence divisor/part row n lists the terms of the n-th row of A338156 in nonincreasing order. In other words: row n lists in nonincreasing order the divisors of the terms of the n-th row of A176206. - Omar E. Pol, Jun 16 2022

Examples

			Triangle begins:
  1;
  2,1,1;
  3,2,1,1,1,1;
  4,3,2,2,2,1,1,1,1,1,1,1;
  5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1;
  6,5,4,4,3,3,3,3,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
  ...
For n = 4 the partitions of 4 are [4], [2, 2], [3, 1], [2, 1, 1], [1, 1, 1, 1]. There is only one 4, only one 3, three 2's and seven 1's, so the 4th row of this triangle is [4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1].
On the other hand for n = 4 the 4th row of A176206 is [4, 3, 2, 2, 1, 1, 1] and the divisors of these terms are [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1], [1] the same as the 4th row of A338156. These divisors listed in nonincreasing order give the 4th row of this triangle. - _Omar E. Pol_, Jun 16 2022

Links

Crossrefs

Both column 1 and 2 are A000027.
Row n has length A006128(n).
The sum of row n is A066186(n).
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).
First differs from A036037, A080577, A181317, A237982 and A239512 at a(13) = T(4,3).
Cf. A302247 (mirror).
Cf. A000041, A027550, A176206, A221529, A336812, A338156.

Programs

  • Mathematica
    nrows=10;Array[ReverseSort[Flatten[IntegerPartitions[#]]]&,nrows] (* Paolo Xausa, Jun 16 2022 *)
  • PARI
    row(n) = my(list = List()); forpart(p=n, for (k=1, #p, listput(list, p[k]));); vecsort(Vec(list), , 4); \\ Michel Marcus, Jun 16 2022

A302247 Irregular triangle read by rows in which row n lists all parts of all partitions of n, in nondecreasing order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Omar E. Pol, Apr 05 2018

Keywords

Comments

Also due to the correspondence divisor/part row n lists the terms of the n-th row of A338156 in nondecreasing order. In other words: row n lists in nondecreasing order the divisors of the terms of the n-th row of A176206. - Omar E. Pol, Jun 16 2022

Examples

			Triangle begins:
  1;
  1,1,2;
  1,1,1,1,2,3;
  1,1,1,1,1,1,1,2,2,2,3,4;
  1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
  1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,5,6;
  ...
For n = 4 the partitions of 4 are [4], [2, 2], [3, 1], [2, 1, 1], [1, 1, 1, 1]. There are seven 1's, three 2's, only one 3 and only one 4, so the 4th row of this triangle is [1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4].
On the other hand for n = 4 the 4th row of A176206 is [4, 3, 2, 2, 1, 1, 1] and the divisors of these terms are [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1], [1] the same as the 4th row of A338156. These divisors listed in nondecreasing order give the 4th row of this triangle. - _Omar E. Pol_, Jun 16 2022

Links

Crossrefs

Mirror of A302246.
Row n has length A006128(n).
The sum of row n is A066186(n).
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).
First differs from both A026791 and A080576 at a(17) = T(4,7).
Cf. A000041, A027750, A176206, A221529, A336812, A338156.

Programs

  • Mathematica
    nrows=10; Array[Sort[Flatten[IntegerPartitions[#]]]&,nrows] (* Paolo Xausa, Jun 16 2022 *)
  • PARI
    row(n) = my(list = List()); forpart(p=n, for (k=1, #p, listput(list, p[k]));); vecsort(Vec(list)); \\ Michel Marcus, Jun 16 2022

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

Views

Author

Omar E. Pol, Jul 31 2021

Keywords

Comments

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.

Examples

			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.

Crossrefs

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.

A210955 Triangle read by rows: T(n,k) = total number of parts <= k in the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 5, 5, 6, 5, 6, 7, 7, 8, 7, 11, 13, 14, 14, 15, 11, 14, 16, 17, 18, 18, 19, 15, 23, 26, 29, 30, 31, 31, 32, 22, 29, 35, 37, 39, 40, 41, 41, 42, 30, 45, 51, 56, 59, 61, 62, 63, 63, 64, 42, 57, 67, 72, 76, 78, 80, 81, 82, 82, 83
Offset: 1

Views

Author

Omar E. Pol, May 01 2012

Keywords

Comments

Row n lists the partial sums of row n of triangle A182703.

Examples

			1,
1,   2,
2,   2,  3,
3,   5,  5,  6,
5,   6,  7,  7,  8,
7,  11, 13, 14, 14, 15,
11, 14, 16, 17, 18, 18, 19,
15, 23, 26, 29, 30, 31, 31, 32,
22, 29, 35, 37, 39, 40, 41, 41, 42;

Crossrefs

Column 1 is A000041. Right border gives A138137.
Cf. A135010, A138121, A182703, A206437, A206562, A207031, A207032, A207383, A208476, A210947, A210956.

Formula

T(n,k) = Sum_{j=1..k} A182703(n,j).

Extensions

More terms from Alois P. Heinz, May 25 2013
Showing 1-5 of 5 results.

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