A337209 -id:A337209 - cp's OEIS frontend

A066186 Sum of all parts of all partitions of n.

0, 1, 4, 9, 20, 35, 66, 105, 176, 270, 420, 616, 924, 1313, 1890, 2640, 3696, 5049, 6930, 9310, 12540, 16632, 22044, 28865, 37800, 48950, 63336, 81270, 104104, 132385, 168120, 212102, 267168, 334719, 418540, 520905, 647172, 800569, 988570, 1216215, 1493520

Offset: 0

Wouter Meeussen, Dec 15 2001

Sum of the zeroth moments of all partitions of n.
Also the number of one-element transitions from the integer partitions of n to the partitions of n-1 for labeled parts with the assumption that any part z is composed of labeled elements of amount 1, i.e., z = 1_1 + 1_2 + ... + 1_z. Then one can take from z a single element in z different ways. E.g., for n=3 to n=2 we have A066186(3) = 9 and [111] --> [11], [111] --> [11], [111] --> [11], [12] --> [111], [12] --> [111], [12] --> [2], [3] --> 2, [3] --> 2, [3] --> 2. For the unlabeled case, one can take a single element from z in only one way. Then the number of one-element transitions from the integer partitions of n to the partitions of n-1 is given by A000070. E.g., A000070(3) = 4 and for the transition from n=3 to n=2 one has [111] --> [11], [12] --> [11], [12] --> [2], [3] --> [2]. - Thomas Wieder, May 20 2004
Also sum of all parts of all regions of n (Cf. A206437). - Omar E. Pol, Jan 13 2013
From Omar E. Pol, Jan 19 2021: (Start)
Apart from initial zero this is also as follows:
Convolution of A000203 and A000041.
Convolution of A024916 and A002865.
For n >= 1, a(n) is also the number of cells in a symmetric polycube in which the terraces are the symmetric representation of sigma(k), for k = n..1, (cf. A237593) starting from the base and located at the levels A000041(0)..A000041(n-1) respectively. The polycube looks like a symmetric tower (cf. A221529). A dissection is a three-dimensional spiral whose top view is described in A239660. The growth of the volume of the polycube represents each convolution mentioned above. (End)
From Omar E. Pol, Feb 04 2021: (Start)
a(n) is also the sum of all divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned divisors are also all parts of all partitions of n.
Apart from initial zero this is also the convolution of A340793 and A000070. (End)

			a(3)=9 because the partitions of 3 are: 3, 2+1 and 1+1+1; and (3) + (2+1) + (1+1+1) = 9.
a(4)=20 because A000041(4)=5 and 4*5=20.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
F. G. Garvan, Higher-order spt functions, Adv. Math. 228 (2011), no. 1, 241-265, alternate copy. - From _N. J. A. Sloane_, Jan 02 2013
F. G. Garvan, Higher-order spt functions, arXiv:1008.1207 [math.NT], 2010.
T. J. Osler, A. Hassen and T. R. Chandrupatia, Surprising connections between partitions and divisors, The College Mathematics Journal, Vol. 38. No. 4, Sep. 2007, 278-287 (see p. 287).
Omar E. Pol, Illustration of a(10), prism and tower, each polycube contains 420 cubes.
Omar E. Pol, Illustration of initial terms of A066186 and of A139582 (n>=1)

Cf. A000041, A093694, A000070, A132825, A001787 (same for ordered partitions), A277029, A000203, A221529, A237593, A239660.
First differences give A138879. - Omar E. Pol, Aug 16 2013
Row sums of triangles A138785, A181187, A245099, A337209, A339106, A340423, A340424, A221529, A302246, A338156, A340035, A340056, A340057, A346741. - Omar E. Pol, Aug 02 2021

a066186 = sum . concat . ps 1 where
   ps _ 0 = [[]]
   ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
-- Reinhard Zumkeller, Jul 13 2013

with(combinat): a:= n-> n*numbpart(n): seq(a(n), n=0..50); # Zerinvary Lajos, Apr 25 2007

PartitionsP[ Range[0, 60] ] * Range[0, 60]

a(n)=numbpart(n)*n \\ Charles R Greathouse IV, Mar 10 2012

from sympy import npartitions
def A066186(n): return n*npartitions(n) # Chai Wah Wu, Oct 22 2023

[n*Partitions(n).cardinality() for n in range(41)] # Peter Luschny, Jul 29 2014

a(n) = n * A000041(n). - Omar E. Pol, Oct 10 2011
G.f.: x * (d/dx) Product_{k>=1} 1/(1-x^k), i.e., derivative of g.f. for A000041. - Jon Perry, Mar 17 2004 (adjusted to match the offset by Geoffrey Critzer, Nov 29 2014)
Equals A132825 * [1, 2, 3, ...]. - Gary W. Adamson, Sep 02 2007
a(n) = A066967(n) + A066966(n). - Omar E. Pol, Mar 10 2012
a(n) = A207381(n) + A207382(n). - Omar E. Pol, Mar 13 2012
a(n) = A006128(n) + A196087(n). - Omar E. Pol, Apr 22 2012
a(n) = A220909(n)/2. - Omar E. Pol, Jan 13 2013
a(n) = Sum_{k=1..n} A000203(k)*A000041(n-k), n >= 1. - Omar E. Pol, Jan 20 2013
a(n) = Sum_{k=1..n} k*A036043(n,n-k+1). - L. Edson Jeffery, Aug 03 2013
a(n) = Sum_{k=1..n} A024916(k)*A002865(n-k), n >= 1. - Omar E. Pol, Jul 13 2014
a(n) ~ exp(Pi*sqrt(2*n/3))/(4*sqrt(3)) * (1 - (sqrt(3/2)/Pi + Pi/(24*sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Oct 24 2016
a(n) = Sum_{k=1..n} A340793(k)*A000070(n-k), n >= 1. - Omar E. Pol, Feb 04 2021

a(0) added by Franklin T. Adams-Watters, Jul 28 2014

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

Omar E. Pol, Oct 14 2020

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)

			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.

Paolo Xausa, Table of n, a(n) for n = 1..13185 (rows 1..19 of triangle, flattened)

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.

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 *)

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

A336812 Irregular triangle read by rows T(n,k), n >= 1, k >= 1, in which row n is constructed replacing every term of row n of A336811 with its divisors.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 20 2020

Here we introduce a new type of table which shows the correspondence between divisors and partitions. More precisely the table shows the corresponce between all parts of the last section of the set of partitions of n and all divisors of all terms of the n-th row of A336811, with n >= 1. The mentionded parts and the mentioned divisors are the same numbers (see Example section).

For an equivalent table showing the same kind of correspondence for all partitions of all positive integers see the supersequence A338156.

			Triangle begins:
  [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];
  ...
For n = 6 the 6th row of A336811 is [6, 4, 3, 2, 2, 1, 1] so replacing every term with its divisors we have {[1, 2, 3, 6], [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1]} the same as the 6th row of this triangle.
Also, if the sequence is written as an irregular tetrahedron so the first six slices are:
  -------------
  [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];
  -------------
The above slices appear in the lower zone of the following table which shows the correspondence between the mentioned divisors and the parts of the last section of the set of partitions of the positive integers.
The table is infinite. It is formed by three zones as follows:
The upper zone shows the last section of the set of partitions of every positive integer.
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      |       6       |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
|   |         |     |       |         |           |             |  6            |
| P |         |     |       |         |           |             |  3 3          |
| A |         |     |       |         |           |             |  4 2          |
| R |         |     |       |         |           |             |  2 2 2        |
| T |         |     |       |         |           |  5          |    1          |
| I |         |     |       |         |           |  3 2        |      1        |
| T |         |     |       |         |  4        |    1        |      1        |
| I |         |     |       |         |  2 2      |      1      |        1      |
| O |         |     |       |  3      |    1      |      1      |        1      |
| N |         |     |  2    |    1    |      1    |        1    |          1    |
| S |         |  1  |    1  |      1  |        1  |          1  |            1  |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
.
|---|---------|-----|-------|---------|-----------|-------------|---------------|
|   | A207031 |  1  |  2 1  |  3 1 1  |  6 3 1 1  |  8 3 2 1 1  | 15 8 4 2 1 1  |
| L |         |  |  |  |/|  |  |/|/|  |  |/|/|/|  |  |/|/|/|/|  |  |/|/|/|/|/|  |
| I | A182703 |  1  |  1 1  |  2 0 1  |  3 2 0 1  |  5 1 1 0 1  |  7 4 2 1 0 1  |
| N |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |  * * * * * *  |
| K | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |  1 2 3 4 5 6  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |  = = = = = =  |
|   | A207383 |  1  |  1 2  |  2 0 3  |  3 4 0 4  |  5 2 3 0 5  |  7 8 6 4 0 6  |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
.
|---|---------|-----|-------|---------|-----------|-------------|---------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |  1 2 3     6  |
| D |---------|-----|-------|---------|-----------|-------------|---------------|
| I | A027750 |     |       |  1      |  1 2      |  1   3      |  1 2   4      |
| V |---------|-----|-------|---------|-----------|-------------|---------------|
| I | A027750 |     |       |         |  1        |  1 2        |  1   3        |
| S |---------|-----|-------|---------|-----------|-------------|---------------|
| O | A027750 |     |       |         |           |  1          |  1 2          |
| R | A027750 |     |       |         |           |  1          |  1 2          |
| S |---------|-----|-------|---------|-----------|-------------|---------------|
|   | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
.
Note that every row in the lower zone lists A027750.
The "section" is the simpler substructure of the set of partitions of n that has this property in the three zones.
Also the lower zone for every positive integer can be constructed using the first n terms of A002865. For example: for n = 6 we consider the first 6 terms of A002865 (that is [1, 0, 1, 1, 2, 2]) and then the 6th slice is formed by a block with the divisors of 6, no block with the divisors of 5, one block with the divisors of 4, one block with the divisors of 3, two blocks with the divisors of 2 and two 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 growth step by step of both the prism of partitions and its associated tower since the number of parts in the last section of the set of partitions of n is equal to A138137(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 in the last section of the set of partitions of n is equal to A138879(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.

Paolo Xausa, Table of n, a(n) for n = 1..10206 (rows 1..23 of triangle, flattened)

Companion and subsequence of A338156.
Row lengths give A138137.
Row sums give A138879.
Cf. A000041, A000070, A002260, A002865, A006128, A024916, A027750, A066186, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A187219, A207031, A207038, A207383, A221529, A221530, A221531, A237593, A245095, A221649, A221650, A302246, A302247, A336811, A337209, A339106, A339258, A339278, A339304, A340035, A340061, A350357.

A336812[row_]:=Flatten[Table[ConstantArray[Divisors[row-m],PartitionsP[m]-PartitionsP[m-1]],{m,0,row-1}]];
Array[A336812,10] (* Generates 10 rows *) (* Paolo Xausa, Feb 16 2023 *)

A340035 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(n-m) copies of the divisors of m, with 1 <= m <= n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 26 2020

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

			Triangle begins:
  1;
  1, 1, 2;
  1, 1, 1, 2, 1, 3;
  1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 4;
  1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 4, 1, 5;
  ...
Written as an irregular tetrahedron the first five slices are:
  1;
  --
  1,
  1, 2;
  -----
  1,
  1,
  1, 2
  1, 3;
  -----
  1,
  1,
  1,
  1, 2,
  1, 2,
  1, 3,
  1, 2, 4;
  --------
  1,
  1,
  1,
  1,
  1,
  1, 2,
  1, 2,
  1, 2,
  1, 3,
  1, 3,
  1, 2, 4,
  1, 5;
--------
The slices of the tetrahedron appear in the upper zone of the following table (formed by three zones) which shows the correspondence between divisors and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      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 table is essentially the same table of A340032 but here, in the upper zone, every row is A027750 instead of A127093.
Also the above table is the table of A338156 upside down.
The connection with the tower described in A221529 is as follows (n = 7):
|--------|------------------------|
| Level  |                        |
| in the | 7th slice of divisors  |
| tower  |                        |
|--------|------------------------|
|  11    |   1,                   |
|  10    |   1,                   |
|   9    |   1,                   |
|   8    |   1,                   |
|   7    |   1,                   |
|   6    |   1,                   |
|   5    |   1,                   |
|   4    |   1,                   |
|   3    |   1,                   |
|   2    |   1,                   |
|   1    |   1,                   |
|--------|------------------------|
|   7    |   1, 2,                |
|   6    |   1, 2,                |
|   5    |   1, 2,                |
|   4    |   1, 2,                |
|   3    |   1, 2,                |
|   2    |   1, 2,                |
|   1    |   1, 2,                |
|--------|------------------------|
|   5    |   1,    3,             |
|   4    |   1,    3,             |
|   3    |   1,    3,             |
|   2    |   1,    3,             |      Level
|   1    |   1,    3,             |             _
|--------|------------------------|       11   | |
|   3    |   1, 2,    4,          |       10   | |
|   2    |   1, 2,    4,          |        9   | |
|   1    |   1, 2,    4,          |        8   |_|_
|--------|------------------------|        7   |   |
|   2    |   1,          5,       |        6   |_ _|_
|   1    |   1,          5,       |        5   |   | |
|--------|------------------------|        4   |_ _|_|_
|   1    |   1, 2, 3,       6,    |        3   |_ _ _| |_
|--------|------------------------|        2   |_ _ _|_ _|_ _
|   1    |   1,                7; |        1   |_ _ _ _|_|_ _|
|--------|------------------------|
             Figure 1.                            Figure 2.
                                                Lateral view
                                                of the tower.
.
                                                _ _ _ _ _ _ _
                                               |_| | | | |   |
                                               |_ _|_| | |   |
                                               |_ _|  _|_|   |
                                               |_ _ _|    _ _|
                                               |_ _ _|  _|
                                               |       |
                                               |_ _ _ _|
.
                                                  Figure 3.
                                                  Top view
                                                of the tower.
.
Figure 1 shows the terms of the 7th row of the triangle arranged as the 7th slice of the tetrahedron. The left hand column (see figure 1) gives the level of the sum of the divisors in the tower (see figures 2 and 3).

Paolo Xausa, Table of n, a(n) for n = 1..17815 (rows 1..20 of triangle, flattened)

Nonzero terms of A340032.
Row lengths give A006128, n >= 1.
Row sums give A066186, n >= 1.
Cf. A000041, A002260, A027750, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A206437, A207031, A207383, A221529, A221530, A221531, A221649, A236104, A237593, A245092, A245095, A221649, A221650, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340011, A340031, A340032, A340056, A340057, A340061.

A340035row[n_]:=Flatten[Array[ConstantArray[Divisors[#],PartitionsP[n-#]]&,n]];
nrows=7;Array[A340035row,nrows] (* Paolo Xausa, Jun 20 2022 *)

A339278 Irregular triangle read by rows T(n,k), (n >= 1, k >= 1), in which the partition number A000041(n-1) is the length of row n and every column k is A000203, the sum of divisors function.

Original entry on oeis.org

1, 3, 4, 1, 7, 3, 1, 6, 4, 3, 1, 1, 12, 7, 4, 3, 3, 1, 1, 8, 6, 7, 4, 4, 3, 3, 1, 1, 1, 1, 15, 12, 6, 7, 7, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1, 13, 8, 12, 6, 6, 7, 7, 4, 4, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 18, 15, 8, 12, 12, 6, 6, 7, 7, 7, 7, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Nov 29 2020

The sum of row n equals A138879(n), the sum of all parts in the last section of the set of partitions of n.

T(n,k) is also the number of cubic cells (or cubes) added at the n-th stage in the k-th level starting from the base in the tower described in A221529, assuming that the tower is an object under construction (see the example). - Omar E. Pol, Jan 20 2022

			Triangle begins:
   1;
   3;
   4,  1;
   7,  3,  1;
   6,  4,  3, 1, 1;
  12,  7,  4, 3, 3, 1, 1;
   8,  6,  7, 4, 4, 3, 3, 1, 1, 1, 1;
  15, 12,  6, 7, 7, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1;
  13,  8, 12, 6, 6, 7, 7, 4, 4, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1;
...
From _Omar E. Pol_, Jan 13 2022: (Start)
Illustration of the first six rows of triangle showing the growth of the symmetric tower described in A221529:
    Level k: 1              2         3        4       5      6     7
Stage
  n   _ _ _ _ _ _ _ _
     |            _  |
  1  |           |_| |
     |_ _ _ _ _ _ _ _|
     |          _    |
     |         | |_  |
  2  |         |_ _| |
     |_ _ _ _ _ _ _ _|_ _ _ _ _ _
     |        _      |        _  |
     |       | |     |       |_| |
  3  |       |_|_ _  |           |
     |         |_ _| |           |
     |_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _
     |      _        |      _    |      _  |
     |     | |       |     | |_  |     |_| |
  4  |     | |_      |     |_ _| |         |
     |     |_  |_ _  |           |         |
     |       |_ _ _| |           |         |
     |_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _|_ _ _ _ _ _ _ _
     |    _          |    _      |    _    |    _  |    _  |
     |   | |         |   | |     |   | |_  |   |_| |   |_| |
     |   | |         |   |_|_ _  |   |_ _| |       |       |
  5  |   |_|_        |     |_ _| |         |       |       |
     |       |_ _ _  |           |         |       |       |
     |       |_ _ _| |           |         |       |       |
     |_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _|_ _ _ _|_ _ _ _|_ _ _ _ _ _
     |  _            |  _        |  _      |  _    |  _    |  _  |  _  |
     | | |           | | |       | | |     | | |_  | | |_  | |_| | |_| |
     | | |           | | |_      | |_|_ _  | |_ _| | |_ _| |     |     |
     | | |_ _        | |_  |_ _  |   |_ _| |       |       |     |     |
  6  | |_    |       |   |_ _ _| |         |       |       |     |     |
     |   |_  |_ _ _  |           |         |       |       |     |     |
     |     |_ _ _ _| |           |         |       |       |     |     |
     |_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _|_ _ _ _|_ _ _ _|_ _ _|_ _ _|
.
Every cell in the diagram of the symmetric representation of sigma represents a cubic cell or cube.
For n = 6 and k = 3 we add four cubes at 6th stage in the third level of the structure of the tower starting from the base so T(6,3) = 4.
For n = 9 another connection with the tower is as follows:
First we take the columns from the above triangle and build a new triangle in which all columns start at row 1 as shown below:
.
   1,  1,  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
   3,  3,  3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3;
   4,  4,  4, 4, 4, 4, 4, 4, 4, 4, 4;
   7,  7,  7, 7, 7, 7, 7;
   6,  6,  6, 6, 6;
  12, 12, 12;
   8,  8;
  15;
  13;
.
Then we rotate the triangle by 90 degrees as shown below:
                                       _
  1;                                  | |
  1;                                  | |
  1;                                  | |
  1;                                  | |
  1;                                  | |
  1;                                  | |
  1;                                  |_|_
  1, 3;                               |   |
  1, 3;                               |   |
  1, 3;                               |   |
  1, 3;                               |_ _|_
  1, 3, 4;                            |   | |
  1, 3, 4;                            |   | |
  1, 3, 4;                            |   | |
  1, 3, 4;                            |_ _|_|_
  1, 3, 4, 7;                         |     | |
  1, 3, 4, 7;                         |_ _ _| |_
  1, 3, 4, 7, 6;                      |     |   |
  1, 3, 4, 7, 6;                      |_ _ _|_ _|_
  1, 3, 4, 7, 6, 12;                  |_ _ _ _| | |_
  1, 3, 4, 7, 6, 12, 8;               |_ _ _ _|_|_ _|_ _
  1, 3, 4, 7, 6, 12, 8, 15; 13;       |_ _ _ _ _|_ _|_ _|
.
                                         Lateral view
                                         of the tower
.                                      _ _ _ _ _ _ _ _ _
                                      |_| | | | | | |   |
                                      |_ _|_| | | | |   |
                                      |_ _|  _|_| | |   |
                                      |_ _ _|    _|_|   |
                                      |_ _ _|  _|    _ _|
                                      |_ _ _ _|     |
                                      |_ _ _ _|  _ _|
                                      |         |
                                      |_ _ _ _ _|
.
                                           Top view
                                         of the tower
.
The sum of the m-th row of the new triangle equals A024916(j) where j is the length of the m-th row, equaling the number of cubic cells in the m-th level of the tower. For example: the last row of triangle has 9 terms and the sum of the last row is 1 + 3 + 4 + 7 + 6 + 12 + 8 + 15 + 13 = A024916(9) = 69, equaling the number of cubes in the base of the tower. (End)

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

Sum of divisors of A336811.
Row n has length A000041(n-1).
Every column gives A000203.
The length of the m-th block in row n is A187219(m), m >= 1.
Row sums give A138879.
Cf. A337209 (another version).
Cf. A272172 (analog for the stepped pyramid described in A245092).
Cf. A024916, A135010, A138121, A138137, A138785, A221529, A235791, A236104, A237270, A237591, A237593, A238442, A338156, A340423, A345023.

A339278[rowmax_]:=Table[Flatten[Table[ConstantArray[DivisorSigma[1,n-m],PartitionsP[m]-PartitionsP[m-1]],{m,0,n-1}]],{n,rowmax}];
A339278[15] (* Generates 15 rows *) (* Paolo Xausa, Feb 17 2023 *)

f(n) = numbpart(n-1);
T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (sigma(n))); my(s=0); while (k <= f(n-1), s++; n--;); sigma(1+s);}
tabf(nn) = {for (n=1, nn, for (k=1, f(n), print1(T(n,k), ", ");); print;);} \\ Michel Marcus, Jan 13 2021

A339278(rowmax)=vector(rowmax,n,concat(vector(n,m,vector(numbpart(m-1)-numbpart(m-2),i,sigma(n-m+1)))));
A339278(15) \\ Generates 15 rows \\ Paolo Xausa, Feb 17 2023

a(m) = A000203(A336811(m)).

T(n,k) = A000203(A336811(n,k)).

A339258 Triangle read by rows T(n,k), (n >= 1, k > = 1), in which row n has length A000070(n-1) and every column gives A000005, the number of divisors function.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 1, 3, 2, 2, 2, 1, 1, 1, 2, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 4, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 4, 2, 2, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 2, 4, 4, 2, 2, 2, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Omar E. Pol, Nov 29 2020

Conjecture: the sum of row n equals A006128(n), the total number of parts in all partitions of n.

			Triangle begins:
  1;
  2, 1;
  2, 2, 1, 1;
  3, 2, 2, 2, 1, 1, 1;
  2, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1;
  4, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
  2, 4, 2, 2, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, ...
  ...

Paolo Xausa, Table of n, a(n) for n = 1..10980 (rows 1..21 of the triangle, flattened)

Row sums give A006128 (conjectured).

Cf. A000005, A000041, A176206, A221530, A221531, A337209.

A339258row[n_]:=Flatten[Table[ConstantArray[DivisorSigma[0,n-m],PartitionsP[m]],{m,0,n-1}]];Array[A339258row,10] (* Paolo Xausa, Sep 02 2023 *)

f(n) = sum(k=0, n-1, numbpart(k));
T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (numdiv(n))); my(s=0); while (k <= f(n-1), s++; n--;); numdiv(1+s);}
tabf(nn) = {for (n=1, nn, for (k=1, f(n), print1(T(n,k), ", ");); print;);} \\ Michel Marcus, Jan 13 2021

T(n,k) = A000005(A176206(n,k)).

A340031 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the j-th row of triangle A127093, where j = n - m + 1 and 1 <= m <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 0, 3, 1, 2, 1, 1, 1, 2, 0, 4, 1, 0, 3, 1, 2, 1, 2, 1, 1, 1, 1, 0, 0, 0, 5, 1, 2, 0, 4, 1, 0, 3, 1, 0, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 5, 1, 2, 0, 4, 1, 2, 0, 4, 1, 0, 3, 1, 0, 3, 1, 0, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Dec 26 2020

Another version of A338156 which is the main sequence with further information about the correspondence divisor/part.

			Triangle begins:
[1];
[1,2],      [1];
[1,0,3],    [1,2],    [1],    [1];
[1,2,0,4],  [1,0,3],  [1,2],  [1,2],  [1],  [1],  [1];
[1,0,0,0,5],[1,2,0,4],[1,0,3],[1,0,3],[1,2],[1,2],[1,2],[1],[1],[1],[1],[1];
[...
Written as an irregular tetrahedron the first five slices are:
[1],
-------
[1, 2],
[1],
----------
[1, 0, 3],
[1, 2],
[1],
[1];
-------------
[1, 2, 0, 4],
[1, 0, 3],
[1, 2],
[1, 2],
[1],
[1],
[1];
----------------
[1, 0, 0, 0, 5],
[1, 2, 0, 4],
[1, 0, 3],
[1, 0, 3],
[1, 2],
[1, 2],
[1, 2],
[1],
[1],
[1],
[1],
[1];
.
The following table formed by three zones shows the correspondence between divisors 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  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
|   |---------|-----|-------|---------|-----------|-------------|
| D | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| I | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| V |---------|-----|-------|---------|-----------|-------------|
| I | A127093 |     |       |         |  1        |  1 2        |
| S | A127093 |     |       |         |  1        |  1 2        |
| O | A127093 |     |       |         |  1        |  1 2        |
| R |---------|-----|-------|---------|-----------|-------------|
| S | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|---|---------|-----|-------|---------|-----------|-------------|
.
The table is essentially the same table of A338156 but here, in the lower zone, every row is A127093 instead of A027750.
.

Paolo Xausa, Table of n, a(n) for n = 1..11552 (rows 1..17 of the triangle, flattened)

Row sums give A066186.
Nonzero terms gives A338156.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A221649, A221650, A237593, A245095, A302246, A302247, A336811, A337209,  A339106, A339258, A339278, A339304, A340011, A340032, A340035, A340061.

A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
A340031row[n_]:=Flatten[Table[ConstantArray[A127093row[n-m+1],PartitionsP[m-1]],{m,n}]];
Array[A340031row,7] (* Paolo Xausa, Sep 28 2023 *)

A340011 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the j-th row of triangle A127093 but with every term multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 0, 3, 1, 2, 2, 1, 2, 0, 4, 1, 0, 3, 2, 4, 3, 1, 0, 0, 0, 5, 1, 2, 0, 4, 2, 0, 6, 3, 6, 5, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 5, 2, 4, 0, 8, 3, 0, 9, 5, 10, 7, 1, 0, 0, 0, 0, 0, 7, 1, 2, 3, 0, 0, 6, 2, 0, 0, 0, 10, 3, 6, 0, 12, 5, 0, 15, 7, 14, 11, 1, 2, 0, 4, 0, 0, 0, 8

Offset: 1

Omar E. Pol, Dec 26 2020

This triangle is a condensed version of the more irregular triangle A340031.

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

			Triangle begins:
[1];
[1, 2],          [1];
[1, 0, 3],       [1, 2],       [2];
[1, 2, 0, 4],    [1, 0, 3],    [2, 4],    [3];
[1, 0, 0, 0, 5], [1, 2, 0, 4], [2, 0, 6], [3, 6], [5];
[...
Row sums give A066186.
Written as an irregular tetrahedron the first five slices are:
--
1;
-----
1, 2,
1;
--------
1, 0, 3,
1, 2,
2;
-----------
1, 2, 0, 4,
1, 0, 3,
2, 4,
3;
--------------
1, 0, 0, 0, 5,
1, 2, 0, 4,
2, 0, 6,
3, 6,
5;
--------------
Row sums give A339106.
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  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
|   |---------|-----|-------|---------|-----------|-------------|
| D | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| I | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| V |---------|-----|-------|---------|-----------|-------------|
| I | A127093 |     |       |         |  1        |  1 2        |
| S | A127093 |     |       |         |  1        |  1 2        |
| O | A127093 |     |       |         |  1        |  1 2        |
| R |---------|-----|-------|---------|-----------|-------------|
| S | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
| C | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
| O |    -    |     |       |  2      |  2 4      |  2 0 6      |
| N |    -    |     |       |         |  3        |  3 6        |
| D |    -    |     |       |         |           |  5          |
|---|---------|-----|-------|---------|-----------|-------------|
.
This lower zone of the table is a condensed version of the "divisors" zone.

Paolo Xausa, Table of n, a(n) for n = 1..11480 (rows 1..40 of the triangle, flattened)

Row sums give A066186.
Nonzero terms give A340056.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A221649, A221650, A237593, A245095, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340031, A340032, A340035, A340061.

A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
A340011row[n_]:=Flatten[Table[A127093row[n-m+1]PartitionsP[m-1],{m,n}]];
Array[A340011row,10] (* Paolo Xausa, Sep 28 2023 *)

A340032 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(n-m) copies of the row m of triangle A127093, with 1 <= m <= n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 0, 3, 1, 1, 1, 1, 2, 1, 2, 1, 0, 3, 1, 2, 0, 4, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 0, 3, 1, 0, 3, 1, 2, 0, 4, 1, 0, 0, 0, 5, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 0, 3, 1, 0, 3, 1, 0, 3, 1, 2, 0, 4, 1, 2, 0, 4, 1, 0, 0, 0, 5, 1, 2, 3, 0, 0, 6

Offset: 1

Omar E. Pol, Dec 26 2020

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

			Triangle begins:
  1;
  1, 1, 2;
  1, 1, 1, 2, 1, 0, 3;
  1, 1, 1, 1, 2, 1, 2, 1, 0, 3, 1, 2, 0, 4;
  1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 0, 3, 1, 0, 3, 1, 2, 0, 4, 1, 0, 0, 0, 5;
  ...
Written as an irregular tetrahedron the first five slices are:
  1;
  --
  1,
  1, 2;
  -----
  1,
  1,
  1, 2,
  1, 0, 3;
  --------
  1,
  1,
  1,
  1, 2,
  1, 2,
  1, 0, 3,
  1, 2, 0, 4;
  -----------
  1,
  1,
  1,
  1,
  1,
  1, 2,
  1, 2,
  1, 2,
  1, 0, 3,
  1, 0, 3,
  1, 2, 0, 4,
  1, 0, 0, 0, 5;
  --------------
  ...
The slices of the tetrahedron appear in the upper zone of the following table (formed by three zones) which shows the correspondence between divisors and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
| D | A127093 |     |       |         |           |  1          |
| I |---------|-----|-------|---------|-----------|-------------|
| V | A127093 |     |       |         |  1        |  1 2        |
| I | A127093 |     |       |         |  1        |  1 2        |
| S | A127093 |     |       |         |  1        |  1 2        |
| O |---------|-----|-------|---------|-----------|-------------|
| R | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| S | A127093 |     |       |  1      |  1 2      |  1 0 3      |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 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 table is essentially the same table of A340035 but here, in the upper zone, every row is A127093 instead of A027750.
Also the above table is the table of A340031 upside down.

Paolo Xausa, Table of n, a(n) for n = 1..11552 (rows 1..17 of the triangle, flattened)

Row sums give A066186.
Nonzero terms gives A340035.
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, A337209, A338156, A339106, A339258, A339278, A339304, A340031, A340061.

A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
A340032row[n_]:=Flatten[Table[ConstantArray[A127093row[m],PartitionsP[n-m]],{m,n}]];
Array[A340032row,7] (* Paolo Xausa, Sep 28 2023 *)

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.

Original entry on oeis.org

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 *)

cp's OEIS Frontend

A066186 Sum of all parts of all partitions of n.

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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.

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A336812 Irregular triangle read by rows T(n,k), n >= 1, k >= 1, in which row n is constructed replacing every term of row n of A336811 with its divisors.

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A340035 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(n-m) copies of the divisors of m, with 1 <= m <= n.

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A339278 Irregular triangle read by rows T(n,k), (n >= 1, k >= 1), in which the partition number A000041(n-1) is the length of row n and every column k is A000203, the sum of divisors function.

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A339258 Triangle read by rows T(n,k), (n >= 1, k > = 1), in which row n has length A000070(n-1) and every column gives A000005, the number of divisors function.

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A340031 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the j-th row of triangle A127093, where j = n - m + 1 and 1 <= m <= n.

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