A206437 -id:A206437 - cp's OEIS frontend

A186114 Triangle of regions and partitions of integers (see Comments lines for definition).

1, 1, 2, 1, 1, 3, 0, 0, 0, 2, 1, 1, 1, 2, 4, 0, 0, 0, 0, 0, 3, 1, 1, 1, 1, 1, 2, 5, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 7

Offset: 1

Omar E. Pol, Aug 08 2011

Let r = T(n,k) be a record in the sequence. The consecutive records "r" are the natural numbers A000027. Consider the first n rows; the triangle T(n,k) has the property that the columns, without the zeros, from k..1, are also the partitions of r in juxtaposed reverse-lexicographical order, so k is also A000041(r), the number of partitions of r. Note that a record r is always the final term of a row if such row contains 1’s. The number of positive integer a(1)..r is A006128(r). The sums a(1)..r is A066186(r). Here the set of positive integers in every row (from 1 to n) is called a “region” of r. The number of regions of r equals the number of partitions of r. If T(n,1) = 1 then the row n is formed by the smallest parts, in nondecreasing order, of all partitions of T(n,n).

			Triangle begins:
1,
1, 2,
1, 1, 3,
0, 0, 0, 2,
1, 1, 1, 2, 4,
0, 0, 0, 0, 0, 3,
1, 1, 1, 1, 1, 2, 5,
0, 0, 0, 0, 0, 0, 0, 2,
0, 0, 0, 0, 0, 0, 0, 2, 4,
0, 0, 0, 0, 0, 0, 0, 0, 0, 3,
1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 6
...
The row n = 11 contains the 6th record in the sequence:  a(66) = T(11,11) = 6, then consider the first 11 rows of triangle. Note that the columns, from k = 11..1, without the zeros, are also the 11 partitions of 6 in juxtaposed reverse-lexicographical order: [6], [3, 3], [4, 2], [2, 2, 2], [5, 1], [3, 2, 1], [4, 1, 1], [2, 2, 1, 1], [3, 1, 1, 1], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1]. See A026792.

Robert Price, Table of n, a(n) for n = 1..196878, rows 1-627.
Omar E. Pol, Illustration of the seven regions of 5

Mirror of triangle A193870. Column 1 gives A167392. Right diagonal gives A141285.

Cf. A000041, A135010, A138121, A183152, A186412, A187219, A194436-A194439, A194446-A194448, A206437.

A206437 = Cases[Import["https://oeis.org/A206437/b206437.txt",
     "Table"], {, }][[All, 2]];
A194446 = Cases[Import["https://oeis.org/A194446/b194446.txt",
    "Table"], {, }][[All, 2]];
f[n_] := Module[{v},
   v = Take[A206437, A194446[[n]]];
   A206437 = Drop[A206437, A194446[[n]]];
   Reverse[PadRight[v, n]]];
Table[f[n], {n, PartitionsP[20]}]  // Flatten (* Robert Price, Apr 26 2020 *)

T(n,1) = A167392(n).

T(n,k) = A141285(n), if k = n.

A193870 Triangle of regions and partitions of integers (see Comments lines for definition).

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 2, 0, 0, 0, 4, 2, 1, 1, 1, 3, 0, 0, 0, 0, 0, 5, 2, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Aug 07 2011

Triangle T(n,k) read by rows in which, from rows 1..n, if r = T(n,k) is a record in the sequence then the set of positive integers in every row (from 1 to n) is called a “region” of r. Note that n, the number of regions of r is also the number of partitions of r. The consecutive records "r" are the natural numbers A000027. The triangle has the property that, for rows n..1, the diagonals (without the zeros) are also the partitions of r, in juxtaposed reverse-lexicographical order. Note that a record "r" is the initial term of a row if such row contains 1’s. If T(n,k) is a record in the sequence then A000041(T(n,k)) = n. Note that if T(n,k) < 2 is not the last term of the row n then T(n,k+1) = T(n,k). The union of the rows that contain 1's gives A182715.

			Triangle begins:
1,
2, 1,
3, 1, 1,
2, 0, 0, 0,
4, 2, 1, 1, 1,
3, 0, 0, 0, 0, 0,
5, 2, 1, 1, 1, 1, 1,
2, 0, 0, 0, 0, 0, 0, 0,
4, 2, 0, 0, 0, 0, 0, 0, 0,
3, 0, 0, 0, 0, 0, 0, 0, 0, 0,
6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1,
…
For n = 11 note that the row n contains the 6th record in the sequence: T(11,1) = a(56) = 6, then consider the first 11 rows of triangle. Note that the diagonals d, from d = n..1, without the zeros, are also the partitions of 6 in juxtaposed reverse-lexicographical order: [6], [3, 3], [4, 2], [2, 2, 2], [5, 1], [3, 2, 1], [4, 1, 1], [2, 2, 1, 1], [3, 1, 1, 1], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1]. See A026792.

Robert Price, Table of n, a(n) for n = 1..196878, rows 1-627.
Omar E. Pol, Illustration of the seven regions of 5

Mirror of triangle A186114. Column 1 gives A141285. Right diagonal gives A167392.

Cf. A046746, A135010, A138121, A182699, A182709, A183152, A186412, A187219, A194436-A194439, A194446-A194448, A206437.

A206437 = Cases[Import["https://oeis.org/A206437/b206437.txt",
     "Table"], {, }][[All, 2]];
A194446 = Cases[Import["https://oeis.org/A194446/b194446.txt",
     "Table"], {, }][[All, 2]];f[n_] := Module[{v},
   v = Take[A206437, A194446[[n]]];
   A206437 = Drop[A206437, A194446[[n]]];
   PadRight[v, n]];
Table[f[n], {n, PartitionsP[20]}] // Flatten (* Robert Price, Apr 26 2020 *)

T(n,1) = A141285(n).

T(n,k) = A167392(n), if k = n.

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

A194447 Rank of the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

Original entry on oeis.org

0, 0, 0, 1, -1, 2, -2, 1, 2, 2, -5, 2, 3, 3, -8, 1, 2, 2, 2, 4, 3, -14, 2, 3, 3, 3, 2, 4, 4, -21, 1, 2, 2, 2, 4, 3, 1, 3, 5, 5, 4, -32, 2, 3, 3, 3, 2, 4, 4, 1, 4, 3, 5, 6, 5, -45, 1, 2, 2, 2, 4, 3, 1, 3, 5, 5, 4, -2, 2, 4, 4, 5, 3, 6, 6, 5, -65

Offset: 1

Omar E. Pol, Dec 04 2011

Here the rank of a "region" is defined to be the largest part minus the number of parts (the same idea as the Dyson's rank of a partition).
Also triangle read by rows: T(j,k) = rank of the k-th region of the last section of the set of partitions of j.
The sum of every row is equal to zero.
Note that in some rows there are several negative terms. - Omar E. Pol, Oct 27 2012
For the definition of "region" see A206437. See also A225600 and A225610. - Omar E. Pol, Aug 12 2013

			In the triangle T(j,k) for j = 6 the number of regions in the last section of the set of partitions of 6 is equal to 4. The first region given by [2] has rank 2-1 = 1. The second region given by [4,2] has rank 4-2 = 2. The third region given by [3] has rank 3-1 = 2. The fourth region given by [6,3,2,2,1,1,1,1,1,1,1] has rank 6-11 = -5 (see below):
From _Omar E. Pol_, Aug 12 2013: (Start)
---------------------------------------------------------
.    Regions       Illustration of ranks of the regions
---------------------------------------------------------
.    For J=6        k=1     k=2      k=3        k=4
.  _ _ _ _ _ _                              _ _ _ _ _ _
. |_ _ _      |                     _ _ _   .          |
. |_ _ _|_    |           _ _ _ _   * * .|    .        |
. |_ _    |   |     _ _   * * .  |              .      |
. |_ _|_ _|_  |     * .|        .|                .    |
.           | |                                     .  |
.           | |                                       .|
.           | |                                       *|
.           | |                                       *|
.           | |                                       *|
.           | |                                       *|
.           |_|                                       *|
.
So row 6 lists:     1       2         2              -5
(End)
Written as a triangle begins:
0;
0;
0;
1,-1;
2,-2;
1,2,2,-5;
2,3,3,-8;
1,2,2,2,4,3,-14;
2,3,3,3,2,4,4,-21;
1,2,2,2,4,3,1,3,5,5,4,-32;
2,3,3,3,2,4,4,1,4,3,5,6,5,-45;
1,2,2,2,4,3,1,3,5,5,4,-2,2,4,4,5,3,6,6,5,-65;
2,3,3,3,2,4,4,1,4,3,5,6,5,-3,3,5,5,4,5,4,7,7,6,-88;

Row j has length A187219(j). The absolute value of the last term of row j is A000094(j+1). Row sums give A000004.

Cf. A000041, A002865, A135010, A138121, A138137, A138879, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194446, A206437.

a(n) = A141285(n) - A194446(n). - Omar E. Pol, Dec 05 2011

A040051 Parity of partition function A000041.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1

Offset: 0

N. J. A. Sloane

From M. V. Subbarao (m.v.subbarao(AT)ualberta.ca), Sep 05 2003: (Start)
Essentially this same question was raised by Ramanujan in a letter to P. A. MacMahon around 1920 (see page 1087, MacMahon's Collected Papers). With the help of Jacobi's triple product identity, MacMahon showed that p(1000) is odd (as he says, with five minutes work - there were no computers those days).
Now we know that among the first ten million values of p(n) 5002137 of them are odd. It is conjectured (T. R. Parkin and D. Shanks) that p(n) is equally often even and odd. Lower bound estimates for the number of times p(n) is even among the first N values of p(n) for any given N are known (Scott Ahlgren; and Nicolas, Rusza and Sárközy among others).
Earlier this year a remarkable result was proved by Boylan and Ahlgren (AMS ABSTRACT # 987-11-82) which says that beyond the three eighty-year old Ramanujan congruences - namely, p(5n+4), p(7n+5) and p(11n +6) being divisible respectively by 5,7 and 11 - there are no other simple congruences of this kind.
My 1966 conjecture that in every arithmetic progression r (mod s) for arbitrary integral r and s, there are infinitely many integers n for which p(n) is odd - with a similar statement for p(n) even - was proved for the even case by Ken Ono (1996) and for the odd case for all s up to 10^5 and for all s which are powers of 2 by Bolyan and Ono, 2002.
(End)
a(n) is also the parity of the trace Tr(n) = A183011(n), the numerator of the Bruinier-Ono formula for the partition function, if n >= 1. - Omar E. Pol, Mar 14 2012
Consider the diagram of the regions of n (see A206437). Then, in each odd-indexed region of n, fill each part of size k with k 1's. Then, in each even-indexed region of n, fill each part of size k with k 0's. The successive digits of row 1 of the diagram give the first n elements of this sequence, if n >= 1. - Omar E. Pol, May 02 2012

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K. Ono, Parity of the partition function, Electron. Res. Announc. AMS, Vol. 1, 1995, pp. 35-42; MR 96d:11108.
Ivars Peterson, Ken Ono's and Nicholas Eriksson's work

Cf. A000041, A071640, A086144, A190938.

import Data.Bits (xor)
a040051 n = p 1 n :: Int where
   p _ 0 = 1
   p k m | k <= m = p k (m - k) `xor` p (k+1) m | k > m  = 0
-- Reinhard Zumkeller, Nov 15 2011

Table[ Mod[ PartitionsP@ n, 2], {n, 105}] (* Robert G. Wilson v, Mar 25 2011 *)

PARI
```
a(n)=if(n<0, 0, numbpart(n)%2)
```

a(n)=if(n<0, 0, polcoeff(1/eta(x+x*O(x^n)), n)%2)

a(n)=if(n<10^9, return(numbpart(n)%2)); my(r=n%4, u=select(k->k^2%32==8*r+1,[1..31]), st=u[1], m=n\4, s); u=[u[2]-u[1],u[3]-u[2],u[4]-u[3],u[1]+32-u[4]]; forstep(t=[1,3,7,5][r+1], sqrtint(32*m-1), u, k=t^2>>5; if(a(m-k), s++)); s%2 \\ Merca's algorithm, switching to direct computation for n less than 10^9. Very time-consuming but low memory use. - Charles R Greathouse IV, Jan 24 2018

from sympy import npartitions
def a(n): return npartitions(n)%2 # Indranil Ghosh, May 25 2017

a(n) = pp(n, 1), with Boolean pp(n, k) = if kReinhard Zumkeller, Sep 04 2003
a(n) = Pm(n,1) with Pm(n,k) = if kReinhard Zumkeller, Jun 09 2009
a(n) = A000035(A000041(n)). - Omar E. Pol, Aug 05 2013
a(n) = A000035(A000025(n)). - John M. Campbell, Jun 29 2016

A186412 Sum of all parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

Original entry on oeis.org

1, 3, 5, 2, 9, 3, 12, 2, 6, 3, 20, 3, 7, 4, 25, 2, 6, 3, 13, 5, 4, 38, 3, 7, 4, 14, 3, 9, 5, 49, 2, 6, 3, 13, 5, 4, 23, 4, 10, 6, 5, 69, 3, 7, 4, 14, 3, 9, 5, 27, 5, 4, 15, 7, 6, 87, 2, 6, 3, 13, 5, 4, 23, 4, 10, 6, 5, 39, 3, 9, 5, 19, 4, 12, 7, 6, 123

Offset: 1

Omar E. Pol, Aug 12 2011

Also triangle read by rows: T(j,k) = sum of all parts in the k-th region of the last section of the set of partitions of j. See Example section. For more information see A135010. - Omar E. Pol, Nov 26 2011

For the definition of "region" see A206437. - Omar E. Pol, Aug 19 2013

			Contribution from Omar E. Pol, Nov 26 2011 (Start):
Written as a triangle:
1;
3;
5;
2,9;
3,12;
2,6,3,20;
3,7,4,25;
2,6,3,13,5,4,38;
3,7,4,14,3,9,5,49;
2,6,3,13,5,4,23,4,10,6,5,69;
3,7,4,14,3,9,5,27,5,4,15,7,6,87;
2,6,3,13,5,4,23,4,10,6,5,39,3,9,5,19,4,12,7,6,123;
(End)
From _Omar E. Pol_, Aug 18 2013: (Start)
Illustration of initial terms (first seven regions):
.                                             _ _ _ _ _
.                                     _ _ _  |_ _ _ _ _|
.                           _ _ _ _  |_ _ _|       |_ _|
.                     _ _  |_ _ _ _|                 |_|
.             _ _ _  |_ _|     |_ _|                 |_|
.       _ _  |_ _ _|             |_|                 |_|
.   _  |_ _|     |_|             |_|                 |_|
.  |_|   |_|     |_|             |_|                 |_|
.
.   1     3       5     2         9       3          12
.
(End)

Row sums of triangle A186114 and of triangle A193870.
Row j has length A187219(j).
Row sums give A138879.
Right border gives A046746, j >= 1.
Records give A046746, j >= 1.
Partial sums give A182244.
Cf. A000041, A002865, A066186, A135010, A138121, A138879, A194436, A194437, A194438, A194439, A194446, A194447.

lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
A186412 = {}; l = {};
For[j = 1, j <= 50, j++,
  mx = Max@lex[j][[j]]; AppendTo[l, mx];
  For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
  AppendTo[A186412, Total@Take[Reverse[First /@ lex[mx]], j - i]];
  ];
A186412  (* Robert Price, Jul 25 2020 *)

a(A000041(n)) = A046746(n).

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 *)

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

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

A225600 Toothpick sequence related to integer partitions (see Comments lines for definition).

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 12, 14, 15, 19, 24, 27, 28, 33, 40, 42, 43, 47, 49, 52, 53, 59, 70, 73, 74, 79, 81, 85, 86, 93, 108, 110, 111, 115, 117, 120, 121, 127, 131, 136, 137, 141, 142, 150, 172, 175, 176, 181, 183, 187, 188, 195, 199, 202, 203, 209, 211, 216, 217, 226, 256

Offset: 0

Omar E. Pol, Jul 28 2013

nonn

This infinite toothpick structure is a minimalist diagram of regions of the set of partitions of all positive integers. For the definition of "region" see A206437. The sequence shows the growth of the diagram as a cellular automaton in which the "input" is A141285 and the "output” is A194446.
To define the sequence we use the following rules:
We start in the first quadrant of the square grid with no toothpicks.
If n is odd we place A141285((n+1)/2) toothpicks of length 1 connected by their endpoints in horizontal direction starting from the grid point (0, (n+1)/2).
If n is even we place toothpicks of length 1 connected by their endpoints in vertical direction starting from the exposed toothpick endpoint downward up to touch the structure or up to touch the x-axis. In this case the number of toothpicks added in vertical direction is equal to A194446(n/2).
The sequence gives the number of toothpicks after n stages. A220517 (the first differences) gives the number of toothpicks added at the n-th stage.
Also the toothpick structure (HV/HHVV/HHHVVV/HHV/HHHHVVVVV...) can be transformed in a Dyck path (UDUUDDUUUDDDUUDUUUUDDDDD...) 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, so the sequence can be represented by the vertices (or the number of steps from the origin) of the Dyck path. Note that the height of the n-th largest peak between two valleys at height 0 is also the partition number A000041(n). See Example section. See also A211978, A220517, A225610.

			For n = 30 the structure has 108 toothpicks, so a(30) = 108.
.                               Diagram of regions
Partitions of 7                 and partitions of 7
.                                   _ _ _ _ _ _ _
7                               15  _ _ _ _      |
4 + 3                               _ _ _ _|_    |
5 + 2                               _ _ _    |   |
3 + 2 + 2                           _ _ _|_ _|_  |
6 + 1                           11  _ _ _      | |
3 + 3 + 1                           _ _ _|_    | |
4 + 2 + 1                           _ _    |   | |
2 + 2 + 2 + 1                       _ _|_ _|_  | |
5 + 1 + 1                        7  _ _ _    | | |
3 + 2 + 1 + 1                       _ _ _|_  | | |
4 + 1 + 1 + 1                    5  _ _    | | | |
2 + 2 + 1 + 1 + 1                   _ _|_  | | | |
3 + 1 + 1 + 1 + 1                3  _ _  | | | | |
2 + 1 + 1 + 1 + 1 + 1            2  _  | | | | | |
1 + 1 + 1 + 1 + 1 + 1 + 1        1   | | | | | | |
.
.                                   1 2 3 4 5 6 7
.
Illustration of initial terms:
.
.                              _ _ _    _ _ _
.                _ _   _ _     _ _      _ _  |
.      _    _    _     _  |    _  |     _  | |
.            |    |     | |     | |      | | |
.
.      1    2     4     6       9        12
.
.
.                          _ _ _ _     _ _ _ _
.      _ _       _ _       _ _         _ _    |
.      _ _ _     _ _|_     _ _|_       _ _|_  |
.      _ _  |    _ _  |    _ _  |      _ _  | |
.      _  | |    _  | |    _  | |      _  | | |
.       | | |     | | |     | | |       | | | |
.
.        14        15         19          24
.
.
.                          _ _ _ _ _    _ _ _ _ _
.    _ _ _      _ _ _      _ _ _        _ _ _    |
.    _ _ _ _    _ _ _|_    _ _ _|_      _ _ _|_  |
.    _ _    |   _ _    |   _ _    |     _ _    | |
.    _ _|_  |   _ _|_  |   _ _|_  |     _ _|_  | |
.    _ _  | |   _ _  | |   _ _  | |     _ _  | | |
.    _  | | |   _  | | |   _  | | |     _  | | | |
.     | | | |    | | | |    | | | |      | | | | |
.
.       27         28         33            40
.
Illustration of initial terms as vertices (or the number of steps from the origin) of a Dyck path:
.
7                                    33
.                                    /\
5                      19           /  \
.                      /\          /    \
3            9        /  \     27 /      \
2       4    /\   14 /    \    /\/        \
1    1  /\  /  \  /\/      \  / 28         \
.    /\/  \/    \/ 15       \/              \
.   0  2   6    12          24              40
.

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. A000041, A006128, A135010, A138137, A139250, A139582, A141285, A186114, A186412, A187219, A194446, A194447, A206437, A207779, A211978, A220517, A225610.

a(A139582(n)) = a(2*A000041(n)) = 2*A006128(n) = A211978(n), n >= 1.

cp's OEIS Frontend

A186114 Triangle of regions and partitions of integers (see Comments lines for definition).

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A193870 Triangle of regions and partitions of integers (see Comments lines for definition).

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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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A194447 Rank of the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

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A040051 Parity of partition function A000041.

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A186412 Sum of all parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

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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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A139582 Twice partition numbers.

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