A066186 -id:A066186 - cp's OEIS frontend

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.

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 *)

A130689 Number of partitions of n such that every part divides the largest part; a(0) = 1.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 11, 16, 19, 26, 28, 41, 43, 56, 65, 82, 88, 115, 122, 155, 174, 209, 225, 283, 305, 363, 402, 477, 514, 622, 666, 783, 858, 990, 1078, 1268, 1362, 1561, 1708, 1958, 2111, 2433, 2613, 2976, 3247, 3652, 3938, 4482, 4821, 5422

Offset: 0

Vladeta Jovovic, Jul 01 2007

First differs from A130714 at a(11) = 28, A130714(11) = 27. - Gus Wiseman, Apr 23 2021

			For n = 6 we have 10 such partitions: [1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 2], [1, 1, 2, 2], [2, 2, 2], [1, 1, 1, 3], [3, 3], [1, 1, 4], [2, 4], [1, 5], [6].
From _Gus Wiseman_, Apr 18 2021: (Start)
The a(1) = 1 through a(8) = 16 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (41)     (33)      (61)       (44)
             (111)  (31)    (221)    (42)      (331)      (62)
                    (211)   (311)    (51)      (421)      (71)
                    (1111)  (2111)   (222)     (511)      (422)
                            (11111)  (411)     (2221)     (611)
                                     (2211)    (4111)     (2222)
                                     (3111)    (22111)    (3311)
                                     (21111)   (31111)    (4211)
                                     (111111)  (211111)   (5111)
                                               (1111111)  (22211)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 1001 terms from Andrew Howroyd)

Cf. A018818, A117086.
The dual version is A083710.
The case without 1's is A339619.
The Heinz numbers of these partitions are the complement of A343337.
The complement is counted by A343341.
The strict case is A343347.
The complement in the strict case is counted by A343377.
A000009 counts strict partitions.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A072233 counts partitions by sum and greatest part.
Cf. A066186, A083711, A097986, A338470, A341450, A343346, A343382.

Table[If[n==0,1,Length[Select[IntegerPartitions[n],FreeQ[#,1]&&And@@IntegerQ/@(Max@@#/#)&]]],{n,0,30}] (* Gus Wiseman, Apr 18 2021 *)

seq(n)={Vec(1 + sum(m=1, n, my(u=divisors(m)); x^m/prod(i=1, #u, 1 - x^u[i] + O(x^(n-m+1)))))} \\ Andrew Howroyd, Apr 17 2021

G.f.: 1 + Sum_{n>0} x^n/Product_{d divides n} (1-x^d).

A339564 Number of ways to choose a distinct factor in a factorization of n (pointed factorizations).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 3, 3, 7, 1, 7, 1, 7, 3, 3, 1, 14, 2, 3, 4, 7, 1, 10, 1, 12, 3, 3, 3, 17, 1, 3, 3, 14, 1, 10, 1, 7, 7, 3, 1, 26, 2, 7, 3, 7, 1, 14, 3, 14, 3, 3, 1, 25, 1, 3, 7, 19, 3, 10, 1, 7, 3, 10, 1, 36, 1, 3, 7, 7, 3, 10, 1, 26, 7, 3

Offset: 1

Gus Wiseman, Apr 10 2021

nonn

			The pointed factorizations of n for n = 2, 4, 6, 8, 12, 24, 30:
  ((2))  ((4))    ((6))    ((8))      ((12))     ((24))       ((30))
         ((2)*2)  ((2)*3)  ((2)*4)    ((2)*6)    ((3)*8)      ((5)*6)
                  (2*(3))  (2*(4))    (2*(6))    (3*(8))      (5*(6))
                           ((2)*2*2)  ((3)*4)    ((4)*6)      ((2)*15)
                                      (3*(4))    (4*(6))      (2*(15))
                                      ((2)*2*3)  ((2)*12)     ((3)*10)
                                      (2*2*(3))  (2*(12))     (3*(10))
                                                 ((2)*2*6)    ((2)*3*5)
                                                 (2*2*(6))    (2*(3)*5)
                                                 ((2)*3*4)    (2*3*(5))
                                                 (2*(3)*4)
                                                 (2*3*(4))
                                                 ((2)*2*2*3)
                                                 (2*2*2*(3))

The additive version is A000070 (strict: A015723).
The unpointed version is A001055 (strict: A045778, ordered: A074206, listed: A162247).
Allowing point (1) gives A057567.
Choosing a position instead of value gives A066637.
The ordered additive version is A336875.
A000005 counts divisors.
A001787 count normal multisets with a selected position.
A001792 counts compositions with a selected position.
A006128 counts partitions with a selected position.
A066186 count strongly normal multisets with a selected position.
A254577 counts ordered factorizations with a selected position.
Cf. A007716, A050336, A281113, A281116, A292886, A293627.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Length[Union[fac]],{fac,facs[n]}],{n,50}]

a(n) = A057567(n) - A001055(n).

a(n) = Sum_{d|n, d>1} A001055(n/d).

A338470 Number of integer partitions of n with no part dividing all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 3, 2, 5, 5, 13, 7, 23, 21, 33, 35, 65, 55, 104, 97, 151, 166, 252, 235, 377, 399, 549, 591, 846, 858, 1237, 1311, 1749, 1934, 2556, 2705, 3659, 3991, 5090, 5608, 7244, 7841, 10086, 11075, 13794, 15420, 19195, 21003, 26240, 29089, 35483

Offset: 0

Gus Wiseman, Mar 23 2021

nonn

Alternative name: Number of integer partitions of n that are empty or have smallest part not dividing all the others.

			The a(5) = 1 through a(12) = 7 partitions (empty column indicated by dot):
  (32)  .  (43)   (53)   (54)    (64)    (65)     (75)
           (52)   (332)  (72)    (73)    (74)     (543)
           (322)         (432)   (433)   (83)     (552)
                         (522)   (532)   (92)     (732)
                         (3222)  (3322)  (443)    (4332)
                                         (533)    (5322)
                                         (542)    (33222)
                                         (632)
                                         (722)
                                         (3332)
                                         (4322)
                                         (5222)
                                         (32222)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The complement is A083710 (strict: A097986).
The strict case is A341450.
The Heinz numbers of these partitions are A342193.
The dual version is A343341.
The case with maximum part not divisible by all the others is A343342.
The case with maximum part divisible by all the others is A343344.
A000005 counts divisors.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A001787 count normal multisets with a selected position.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A167865 counts strict chains of divisors > 1 summing to n.
A276024 counts positive subset sums.
Sequences with similar formulas: A024994, A047966, A047968, A168111.
Cf. A001792, A064391, A064410, A066186, A067824, A083711, A098965, A264401, A339562, A339563.

Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}]
(* Second program: *)
a[n_] := If[n == 0, 1, PartitionsP[n] - Sum[PartitionsP[d-1], {d, Divisors[n]}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 09 2021, after Andrew Howroyd *)

a(n)={numbpart(n) - if(n, sumdiv(n, d, numbpart(d-1)))} \\ Andrew Howroyd, Mar 25 2021

a(n) = A000041(n) - Sum_{d|n} A000041(d-1) for n > 0. - Andrew Howroyd, Mar 25 2021

A138785 Triangle read by rows: T(n,k) is the number of hook lengths equal to k among all hook lengths of all partitions of n (1 <= k <= n).

Original entry on oeis.org

1, 2, 2, 4, 2, 3, 7, 6, 3, 4, 12, 8, 6, 4, 5, 19, 16, 12, 8, 5, 6, 30, 22, 18, 12, 10, 6, 7, 45, 38, 27, 24, 15, 12, 7, 8, 67, 52, 45, 32, 25, 18, 14, 8, 9, 97, 82, 63, 52, 40, 30, 21, 16, 9, 10, 139, 112, 93, 72, 60, 42, 35, 24, 18, 10, 11, 195, 166, 135, 112, 85, 72, 49, 40, 27, 20, 11, 12

Offset: 1

Emeric Deutsch, May 16 2008

T(n,k) is also the sum of all parts of size k in all partitions of n. - Omar E. Pol, Feb 24 2012

T(n,k) is also the sum of all k's that are 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. - Omar E. Pol, Feb 05 2021

			T(4,2) = 6 because for the partitions (4), (3,1), (2,2), (2,1,1), (1,1,1,1) of n=4 the hook length multi-sets are {4,3,2,1}, {4,2,1,1}, {3,2,2,1}, {4,1,2,1}, {4,3,2,1}, respectively, containing altogether six 2's.
Triangle starts:
   1;
   2,  2;
   4,  2,  3;
   7,  6,  3,  4;
  12,  8,  6,  4,  5;
  19, 16, 12,  8,  5,  6;
  30, 22, 18, 12, 10,  6,  7;
  45, 38, 27, 24, 15, 12,  7,  8;
  67, 52, 45, 32, 25, 18, 14,  8, 9;
  97, 82, 63, 52, 40, 30, 21, 16, 9, 10;

Alois P. Heinz, Rows n = 1..141, flattened
R. Bacher and L. Manivel, Hooks and powers of parts in partitions, Sem. Lotharingien de Combinatoire, 47, 2002, B47d.
Guo-Niu Han, An explicit expansion formula for the powers of the Euler product in terms of partition hook lengths, arXiv:0804.1849v3 [math.CO] 9 May 2008 (p. 24).

Row sums yield A066186.

Cf. A000041, A066186, A000070, A066183, A066633.

g:=sum(k*x^k*t^k/((1-x^k)*(product(1-x^m,m=1..50))),k=1..50): gser:= simplify(series(g,x=0,15)): for n to 12 do P[n]:= sort(coeff(gser,x,n)) end do: for n to 12 do seq(coeff(P[n],t,j),j=1..n) end do; # yields sequence in triangular form
# second program:
b:= proc(n, i) option remember; `if`(n=0, [1],
     `if`(i=1, [1, n], (p-> (g-> p(p(b(n, i-1), g),
      [0$i, g[1]]))(`if`(i>n, [0], b(n-i, i))))(
      (f, g)-> zip((x, y)-> x+y, f, g, 0))))
    end:
T:= n-> (l-> seq(l[i+1]*i, i=1..n))(b(n$2)):
seq(T(n), n=1..14);  # Alois P. Heinz, Mar 22 2012

max = 12; s = Series[Sum[k*t^k*x^k/((1 - x^k)*Product[1 - x^m, {m, 1, max}]), {k, 1, max}] , {x, 0, max}, {t, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {t, 0, k}]; Table[t[n, k], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 17 2014 *)
Table[Count[Flatten@IntegerPartitions@n, k]*k, {n, 12}, {k, n}] // Flatten (* Robert Price, Jun 15 2020 *)

T(n,1) = A000070(n-1).
Sum_{k=1..n} k*T(n,k) = A066183(n).
G.f.: Sum(k*t^k*x^k/[(1-x^k)*Product(1-x^m, m=1..infinity)], k=1..infinity).
T(n,k) = k*A066633(n,k).
T(n,k) = Sum_{j=1..n} A207383(j,k). - Omar E. Pol, May 02 2012

A139582 Twice partition numbers.

Original entry on oeis.org

2, 2, 4, 6, 10, 14, 22, 30, 44, 60, 84, 112, 154, 202, 270, 352, 462, 594, 770, 980, 1254, 1584, 2004, 2510, 3150, 3916, 4872, 6020, 7436, 9130, 11208, 13684, 16698, 20286, 24620, 29766, 35954, 43274, 52030, 62370, 74676, 89166, 106348, 126522, 150350, 178268, 211116, 249508, 294546, 347050, 408452

Offset: 0

Omar E. Pol, May 14 2008

Except for the first term the number of segments needed to draw (on the infinite square grid) a minimalist diagram of regions and partitions of n. Therefore A000041(n) is also the number of pairs of orthogonal segments (L-shaped) in the same diagram (See links section). For the definition of "regions of n" see A206437. - Omar E. Pol, Oct 29 2012

			The number of partitions of 6 is 11, then a(6) = 2*11 = 22.

V. Modrak, D. Marton, A framework for generating and complexity assessment of assembly supply chains, in Nonlinear Science and Complexity (NSC), 2012 IEEE 4th International Conference on, Date of Conference: 6-11 Aug. 2012; Digital Object Identifier: 10.1109/NSC.2012.6304712. - From _N. J. A. Sloane_, Dec 27 2012
V. Modrak, D. Marton, Development of Metrics and a Complexity Scale for the Topology of Assembly Supply Chains, Entropy 2013, 15, 4285-4299; doi:10.3390/e15104285.
V. Modrak, D. Marton, Approaches to Defining and Measuring Assembly Supply Chain Complexity, Discontinuity and Complexity in Nonlinear Physical Systems, Vol. 6, 2014, pp. 192-213.
V. Modrak, D. Marton, Configuration complexity assessment of convergent supply chain systems, International Journal of General Systems, Volume 43, Issue 5, 2014.
Omar E. Pol, Illustration of initial terms of A139582 (n>=1) and of A066186

Cf. A000041, A135010, A206437, A211026.

Array[2 PartitionsP@# &, 50, 0] (* Robert G. Wilson v, Feb 11 2018 *)

a(n) = 2*numbpart(n); \\ Michel Marcus, Feb 12 2018

a(n) = 2*A000041(n).

More terms from Omar E. Pol, Feb 11 2018

A182709 Sum of the emergent parts of the partitions of n.

Original entry on oeis.org

0, 0, 0, 2, 3, 11, 14, 33, 45, 81, 109, 185, 237, 372, 490, 715, 928, 1326, 1693, 2348, 2998, 4032, 5119, 6795, 8530, 11132, 13952, 17927, 22314, 28417, 35126, 44279, 54532, 68062, 83422, 103427, 126063, 155207, 188506, 230547, 278788, 339223, 408482

Offset: 1

Omar E. Pol, Nov 28 2010, Nov 29 2010

Here the "emergent parts" of the partitions of n are defined to be the parts (with multiplicity) of all the partitions that do not contain "1" as a part, removed by one copy of the smallest part of every partition. Note that these parts are located in the head of the last section of the set of partitions of n. For more information see A182699.

Also total sum of parts of the regions that do not contain 1 as a part in the last section of the set of partitions of n (Cf. A083751, A187219). - Omar E. Pol, Mar 04 2012

			For n=7 the partitions of 7 that do not contain "1" as a part are
7
4 + 3
5 + 2
3 + 2 + 2
Then remove one copy of the smallest part of every partition. The rest are the emergent parts:
.,
4, .
5, .
3, 2, .
The sum of these parts is 4 + 5 + 3 + 2 = 14, so a(7)=14.
For n=10 the illustration in the link shows the location of the emergent parts (colored yellow and green) and the location of the filler parts (colored blue) in the last section of the set of partitions of 10.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Jason Kimberley)
Omar E. Pol, Illustration: How to build the last section of the set of partitions (copy, paste and fill)
Omar E. Pol, Illustration of the shell model of partitions (2D view)

Cf. A000041, A135010, A138121, A138879, A138880, A182699, A182703, A182708, A182740, A182742, A182743.

Row sums of A183152.

b:= proc(n, i) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i<2 then 0
    else b(n, i-1) +b(n-i, i)
      fi
    end:
c:= proc(n, i, k) option remember;
      if n<0 then 0
    elif n=0 then k
    elif i<2 then 0
    else c(n, i-1, k) +c(n-i, i, i)
      fi
    end:
a:= n-> n*b(n, n) - c(n, n, 0):
seq(a(n), n=1..40);  #  Alois P. Heinz, Dec 01 2010

f[n_]:=Total[Flatten[Most/@Select[IntegerPartitions[n],!MemberQ[#,1]&]]]; Table[f[i],{i,50}] (* Harvey P. Dale, Dec 28 2010 *)
b[n_, i_] := b[n, i] = Which[n<0, 0, n==0, 1, i<2, 0, True, b[n, i-1] + b[n - i, i]]; c[n_, i_, k_] := c[n, i, k] = Which[n<0, 0, n==0, k, i<2, 0, True, c[n, i-1, k] + c[n-i, i, i]]; a[n_] := n*b[n, n] - c[n, n, 0]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Oct 08 2015, after Alois P. Heinz *)

a(n) = A138880(n) - A182708(n).
a(n) = A066186(n) - A066186(n-1) - A046746(n) = A138879(n) - A046746(n). - Omar E. Pol, Aug 01 2013
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (12*sqrt(2*n)) * (1 - (3*sqrt(3/2)/Pi + 13*Pi/(24*sqrt(6)))/sqrt(n)). - Vaclav Kotesovec, Jan 03 2019, extended Jul 05 2019

More terms from Alois P. Heinz, Dec 01 2010

A183011 (24n - 1)p(n): traces of partition class polynomials, with a(0) = -1.

Original entry on oeis.org

-1, 23, 94, 213, 475, 833, 1573, 2505, 4202, 6450, 10038, 14728, 22099, 31411, 45225, 63184, 88473, 120879, 165935, 222950, 300333, 398376, 528054, 691505, 905625, 1172842, 1517628, 1947470, 2494778, 3172675, 4029276, 5083606, 6403683, 8023113

Offset: 0

Omar E. Pol, Jan 21 2011

sign

a(n) is also Tr(n), the numerator of the finite algebraic formula for the number of partitions of n, if n >= 1. The formula is  p(n) = Tr(n)/(24*n - 1), n >= 1. See theorem 1.1 of the Bruinier-Ono paper in the link. For the denominators see A183010.
a(n) is also the coefficient of the second term (the trace) in the n-th Bruinier-Ono "partition polynomial" H_n(x), if n >= 1. See the Bruinier-Ono paper, theorem 1.1 and chapter 5 "Examples". For the coefficients of the 4th terms see A187218. - Omar E. Pol, Jul 10 2011
In the Bruinier-Ono-Sutherland paper (Jan 23 2013) partition polynomials are called "partition class polynomials". See also Sutherland's table of Hpart_n(x) in link section. - Omar E. Pol, Feb 20 2013

			1. For n = 6, the number of partitions of 6 is 11, so a(6) = (24*6 - 1)*11 = 143*11 = 1573.
2. For n = 1, in the Bruinier-Ono paper, chapter 5, the first "partition polynomial" is H_1(x) = x^3 - 23*x^2 + (3592/23)*x - 419. The coefficient of the second term (the trace) is 23, so a(1) = 23.
G.f. = -1 + 23*x + 94*x^2 + 213*x^3 + 475*x^4 + 833*x^5 + 1573*x^6 + 2505*x^7 + ...
G.f. = -q^-1 + 23*q^23 + 94*q^47 + 213*q^71 + 475*q^95 + 833*q^119 + 1573*q^143 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
K. Bringmann and K. Ono, An arithmetic formula for the partition function, Proc Am. Math. Soc. 135 (2007), 3507-3515
K. Bringmann and K. Ono, Lifting elliptic cusp forms to Maass forms with an application to partitions, Proc Nat. Acad. Sci. 104 (10) (2007) 3725-3731
J. H. Bruinier and K. Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms
J. H. Bruinier, K. Ono, A. V. Sutherland, Class polynomials for nonholomorphic modular functions, arXiv:1301.5672 [math.NT], 2013-2015.
A. Dabholkar, S. Murthy, D. Zagier, Quantum Black Holes, Wall Crossing, and Mock Modular Forms, arXiv:1208.4074 [hep-th], 2012-2014, p. 46.
A. Folsom, Z. A. Kent and K. Ono, l-adic properties of the partition function, preprint.
A. Folsom, Z. A. Kent and K. Ono, l-adic properties of the partition function, Advances in Mathematics, 229 (2012), pages 1586-1609.
F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences, arXiv:1011.1957 [math.NT], 2010; see (1.5), (2.10).
P. M. Jenkins, Traces of singular moduli, modular forms, and Maass forms
E. Larson and L. Rolen, Integrality properties of the CM-values of certain weak Maass forms, arXiv:1107.4114 [math.NT], 2011.
K. Ono, Congruences for the Andrews spt-function, (see 2.1 Producing modular forms)
A. V. Sutherland, Partition class polynomials, Hpart_n(x), n = 1..770

Positive terms are the partial sums of A183012, also the column 24 of A182729.

Cf. A000041, A008606, A066186, A183006, A183009, A183010, A183054, A187206.

a[ n_] := (24 n - 1) SeriesCoefficient[ 1 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Jun 26 2017 *)

{a(n) = if( n<0, 0, (24*n - 1) * numbpart(n))}; /* Michael Somos, Aug 28 2013 */

a(n) = A183010(n)*A000041(n).
a(n) = 24*A066186(n) - A000041(n) = A183009(n) - A000041(n) = (A008606(n)-1)*A000041(n).
a(n) = 12M_2(n) - p(n) = 24spt(n) + 12N_2(n) - p(n) = 12*A220909(n) - A000041(n) = 24*A092269(n) + 12*A220908(n) - A000041(n), n >= 1. - Omar E. Pol, Feb 17 2013
G.f.: Sum_{k >= 0} a(k) * q^(24*k - 1) = q * d/dq (1/q * Product_{k > 0} 1 / (1 - q^(24*k))). - Michael Somos, Aug 28 2013

A211978 Total number of parts in all partitions of n plus the sum of largest parts of all partitions of n.

Original entry on oeis.org

0, 2, 6, 12, 24, 40, 70, 108, 172, 256, 384, 550, 798, 1112, 1560, 2136, 2926, 3930, 5288, 6996, 9260, 12104, 15798, 20412, 26348, 33702, 43044, 54588, 69090, 86906, 109126, 136270, 169854, 210732, 260924, 321752, 396028, 485624, 594402, 725174, 883092, 1072208

Offset: 0

Omar E. Pol, Jan 03 2013

nonn

Also twice A006128, because the total number of parts in all partitions of n equals the sum of largest parts of all partitions of n. For a proof without words see the illustration of initial terms. Note that the sum of the lengths of all horizontal segments equals the sum of largest parts of all partitions of n. On the other hand, the sum of the lengths of all vertical segments equals the total number of parts of all partition of n. Therefore the sum of lengths of all horizontal segments equals the sum of lengths of all vertical segments.
a(n) is also the sum of the semiperimeters of the Ferrers boards of the partitions of n. Example: a(2)=6; indeed, the Ferrers boards of the partitions [2] and [1,1] of 2 are 2x1 rectangles; the sum of their semiperimeters is 3 + 3 = 6. - Emeric Deutsch, Oct 07 2016
a(n) is also the sum of the semiperimeters of the regions of the set of partitions of n. See the first illustration in the Example section. For more information see A278355. - Omar E. Pol, Nov 23 2016

			Illustration of initial terms as a minimalist diagram of regions of the set of partitions of n, for n = 1..6:
.                                         _ _ _ _ _ _
.                                         _ _ _      |
.                                         _ _ _|_    |
.                                         _ _    |   |
.                             _ _ _ _ _   _ _|_ _|_  |
.                             _ _ _    |  _ _ _    | |
.                   _ _ _ _   _ _ _|_  |  _ _ _|_  | |
.                   _ _    |  _ _    | |  _ _    | | |
.           _ _ _   _ _|_  |  _ _|_  | |  _ _|_  | | |
.     _ _   _ _  |  _ _  | |  _ _  | | |  _ _  | | | |
. _   _  |  _  | |  _  | | |  _  | | | |  _  | | | | |
.  |   | |   | | |   | | | |   | | | | |   | | | | | |
.
. 2    6     12        24         40          70
.
Also using the elements from the diagram we can draw an infinite Dyck path in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. Note that the n-th largest peak between two valleys at height 0 is also the partition number A000041(n) as shown below:
.
11...........................................................
.                                                           /\
.                                                          /  \
.                                                         /    \
7..................................                      /      \
.                                 /\                    /        \
5....................            /  \                /\/          \
.                   /\          /    \          /\  /              \
3..........        /  \        /      \        /  \/                \
2.....    /\      /    \    /\/        \      /                      \
1..  /\  /  \  /\/      \  /            \  /\/                        \
0 /\/  \/    \/          \/              \/                            \
. 0,2,  6,   12,         24,             40,                          70...
.

Michael De Vlieger, Table of n, a(n) for n = 0..3000
Toufik Mansour, Armend Sh. Shabani, Enumerations on bargraphs, Discrete Math. Lett. (2019) Vol. 2, 65-94.
Omar E. Pol, Visualization of regions in a diagram for A006128
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139582 is twice A000041. A220909 is twice A066186.

Cf. A006128, A135010, A141285, A186114, A193870, A187219, A194446, A194447, A206437, A211026, A220517, A225600, A278355.

Q := sum(x^j/(1-x^j), j = 1 .. i): R := product(1-x^j, j = 1 .. i): g := sum(x^i*(1+i+Q)/R, i = 1 .. 100): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 41); # Emeric Deutsch, Oct 07 2016

Array[2 Sum[DivisorSigma[0, m] PartitionsP[# - m], {m, #}] &, 42, 0] (* Michael De Vlieger, Mar 20 2020 *)

a(n) = 2*A006128(n).
a(n) = A225600(2*A000041(n)) = A225600(A139582(n)), n >= 1.
a(n) = (Sum_{m=1..p(n)} A194446(m)) + (Sum_{m=1..p(n)} A141285(m)) = 2*Sum_{m=1..p(n)} A194446(m) = 2*Sum_{m=1..p(n)} A141285(m), where p(n) = A000041(n), n >= 1.
The trivariate g.f. G(t,s,x) of the partitions of a nonnegative integer relative to weight (marked by x), number of parts (marked by t), and largest part (marked by s) is G(t,s,x) = Sum_{i>=1} t*s^i*x^i/product_{j=1..i} (1-tx^j). Setting s = t, we obtain the bivariate g.f. of the partitions relative to weight (marked by x) and semiperimeter of the Ferrers board (marked by t). The g.f. of a(n) is g(x) = Sum_{i>=1} ((x^i*(1 + i + Q(x))/R(x)), where Q(x) = sum_{j=1..i} (x^j/(1 - x^j)) and R(x) = product_{j=1..i}(1-x^j). g(x) has been obtained by setting t = 1 in dG(t,t,x))/dt. - Emeric Deutsch, Oct 07 2016

cp's OEIS Frontend

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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A130689 Number of partitions of n such that every part divides the largest part; a(0) = 1.

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A339564 Number of ways to choose a distinct factor in a factorization of n (pointed factorizations).

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A338470 Number of integer partitions of n with no part dividing all the others.

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A138785 Triangle read by rows: T(n,k) is the number of hook lengths equal to k among all hook lengths of all partitions of n (1 <= k <= n).

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A139582 Twice partition numbers.

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A182709 Sum of the emergent parts of the partitions of n.