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A286001 A table of partitions into consecutive parts (see Comments lines for definition).

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 5, 2, 6, 3, 1, 7, 3, 2, 8, 4, 3, 9, 4, 2, 10, 5, 3, 1, 11, 5, 4, 2, 12, 6, 3, 3, 13, 6, 4, 4, 14, 7, 5, 2, 15, 7, 4, 3, 1, 16, 8, 5, 4, 2, 17, 8, 6, 5, 3, 18, 9, 5, 3, 4, 19, 9, 6, 4, 5, 20, 10, 7, 5, 2, 21, 10, 6, 6, 3, 1, 22, 11, 7, 4, 4, 2, 23, 11, 8, 5, 5, 3, 24, 12, 7, 6, 6, 4, 25, 12, 8, 7, 3, 5
Offset: 1

Views

Author

Omar E. Pol, Apr 30 2017

Keywords

Comments

This is a triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists successive blocks of k consecutive terms, where the m-th block starts with m, m>=1, and the first element of column k is in row k*(k+1)/2.
The partitions of n into consecutive parts are represented from the row n up to row A288529(n) as maximum, but in increasing order, exclusively in the columns where the blocks begin.
More precisely, the partition of n into exactly k consecutive parts (if such partition exists) is represented in the column k from the row n up to row n + k - 1 (see examples).
A288772(n) is the minimum number of rows that are required to represent in this table the partitions of all positive integers <= n into consecutive parts.
A288773(n) is the largest of all positive integers whose partitions into consecutive parts can be totally represented in the first n rows of this table.
A288774(n) is the largest positive integers whose partitions into consecutive parts can be totally represented in the first n rows of this table.
For a theorem related to this table see A286000.

Examples

			Triangle begins:
1;
2;
3,   1;
4,   2;
5,   2;
6,   3,  1;
7,   3,  2;
8,   4,  3;
9,   4,  2;
10,  5,  3,  1;
11,  5,  4,  2;
12,  6,  3,  3;
13,  6,  4,  4;
14,  7,  5,  2;
15,  7,  4,  3,  1;
16,  8,  5,  4,  2;
17,  8,  6,  5,  3;
18,  9,  5,  3,  4;
19,  9,  6,  4,  5;
20, 10,  7,  5,  2;
21, 10,  6,  6,  3,  1;
22, 11,  7,  4,  4,  2;
23, 11,  8,  5,  5,  3;
24, 12,  7,  6,  6,  4;
25, 12,  8,  7,  3,  5;
26, 13,  9,  5,  4,  6;
27, 13,  8,  6,  5,  2;
28, 14,  9,  7,  6,  3,  1;
...
Figures A..G show the location (in the columns of the table) of the partitions of n = 1..7 (respectively) into consecutive parts:
.   ------------------------------------------------------------------------
Fig:   A     B       C         D          E            F             G
.   ------------------------------------------------------------------------
. n:   1     2       3         4          5            6             7
Row ------------------------------------------------------------------------
1   | [1];|  1; |  1;     |  1;    |  1;        |  1;         |  1;        |
2   |     | [2];|  2;     |  2;    |  2;        |  2;         |  2;        |
3   |     |     | [3],[1];|  3,  1;|  3,  1;    |  3,  1;     |  3,  1;    |
4   |     |     |  4 ,[2];| [4], 2;|  4,  2;    |  4,  2;     |  4,  2;    |
5   |     |     |         |        | [5],[2];   |  5,  2;     |  5,  2;    |
6   |     |     |         |        |  6, [3], 3;| [6], 3, [1];|  6,  3,  1;|
7   |     |     |         |        |            |  7,  3, [2];| [7],[3], 2;|
8   |     |     |         |        |            |  8,  4, [3];|  8, [4], 3;|
.   ------------------------------------------------------------------------
Figure F: for n = 6 the partitions of 6 into consecutive parts (but with the parts in increasing order) are [6] and [1, 2, 3]. These partitions have 1 and 3 consecutive parts respectively. On the other hand  we can find the mentioned partitions in the columns 1 and 3 of this table, starting at the row 6.
.
Figures H..K show the location (in the columns of the table) of the partitions of 8..11 (respectively) into consecutive parts:
.    --------------------------------------------------------------------
Fig:        H             I                  J                 K
.    --------------------------------------------------------------------
. n:        8             9                  10                11
Row  --------------------------------------------------------------------
1    |  1;        |  1;            |   1;             |   1;            |
1    |  2;        |  2;            |   2;             |   2;            |
3    |  3,  1;    |  3,  1;        |   3,  1;         |   3,  1;        |
4    |  4,  2;    |  4,  2;        |   4,  2;         |   4,  2;        |
5    |  5,  2;    |  5,  2;        |   5,  2;         |   5,  2;        |
6    |  6,  3,  3;|  6,  3,  1;    |   6,  3,  1;     |   6,  3,  1;    |
7    |  7,  3,  2;|  7,  3,  2;    |   7,  3,  2;     |   7,  3,  2;    |
8    | [8], 4,  1;|  8,  4,  3;    |   8,  4,  3;     |   8,  4,  3;    |
9    |            | [9],[4],[2];   |   9,  4,  2;     |   9,  4,  2;    |
10   |            | 10, [5],[3], 1;| [10], 5,  3, [1];|  10,  5,  3,  1;|
11   |            | 11,  5, [4], 2;|  11,  5,  4, [2];| [11],[5], 4,  2;|
12   |            |                |  12,  6,  3, [3];|  12, [6], 3,  3;|
13   |            |                |  13,  6,  4, [4];|  13,  6,  4,  4;|
.    --------------------------------------------------------------------
Figure J: For n = 10 the partitions of 10 into consecutive parts (but with the parts in increasing order) are [10] and [1, 2, 3, 4]. These partitions have 1 and 4 consecutive parts respectively. On the other hand, we can find the mentioned partitions in the columns 1 and 4 of this table, starting at the row 10.
.
Illustration of initial terms arranged into the diagram of the triangle A237591:
.                                                           _
.                                                         _|1|
.                                                       _|2 _|
.                                                     _|3  |1|
.                                                   _|4   _|2|
.                                                 _|5    |2 _|
.                                               _|6     _|3|1|
.                                             _|7      |3  |2|
.                                           _|8       _|4 _|3|
.                                         _|9        |4  |2 _|
.                                       _|10        _|5  |3|1|
.                                     _|11         |5   _|4|2|
.                                   _|12          _|6  |3  |3|
.                                 _|13           |6    |4 _|4|
.                               _|14            _|7   _|5|2 _|
.                             _|15             |7    |4  |3|1|
.                           _|16              _|8    |5  |4|2|
.                         _|17               |8     _|6 _|5|3|
.                       _|18                _|9    |5  |3  |4|
.                     _|19                 |9      |6  |4 _|5|
.                   _|20                  _|10    _|7  |5|2 _|
.                 _|21                   |10     |6   _|6|3|1|
.               _|22                    _|11     |7  |4  |4|2|
.             _|23                     |11      _|8  |5  |5|3|
.           _|24                      _|12     |7    |6 _|6|4|
.         _|25                       |12       |8   _|7|3  |5|
.       _|26                        _|13      _|9  |5  |4 _|6|
.     _|27                         |13       |8    |6  |5|2 _|
.    |28                           |14       |9    |7  |6|3|1|
...
The number of horizontal line segments in the n-th row of the diagram equals A001227(n), the number of partitions of n into consecutive parts.
.
From _Omar E. Pol_, Dec 15 2020: (Start)
The connection (described step by step) between the triangle of A299765 and the above geometric diagram is as follows:
.
   [1];                                       [1];
   [2];                                       [2];
   [3], [2, 1];                               [3], [2, 1];
   [4];                                       [4];
   [5], [3, 2];                               [5], [3, 2];
   [6], [3, 2, 1];                            [6],         [3, 2, 1];
   [7], [4, 3];                               [7], [4, 3];
   [8];                                       [8];
   [9], [5, 4], [4, 3, 2];                    [9], [5, 4], [4, 3, 2];
.
         Figure 1.                                   Figure 2.
.
We start with the irregular                Then we write the same triangle
triangle of A299765 in which               but ordered in columns where the
row n lists the partitions                 column k lists the partitions of
of n into consecutive parts.               n into k consecutive parts.
.
.   _                                          _
    1|                                        |1
    _                                          _
    2|                                        |2
    _    _ _                                   _      _
    3|   2,1|                                 |3     |1
    _                                          _     |2
    4|                                        |4
    _    _ _                                   _      _
    5|   3,2|                                 |5     |2
    _           _ _ _                          _     |3      _
    6|          3,2,1|                        |6            |1
    _    _ _                                   _      _     |2
    7|   4,3|                                 |7     |3     |3
    _                                          _     |4
    8|                                        |8
    _    _ _    _ _ _                          _      _      _
    9|   5,4|   4,3,2|                        |9     |4     |2
                                                     |5     |3
                                                            |4
.
         Figure 3.                                Figure 4.
.
Then we draw to the right of               Then we rotate each sub-diagram
each partition a vertical                  90 degrees counterclockwise.
toothpick and above each part              Every horizontal toothpick represents
we draw a horizontal toothpick.            the existence of that partition.
.                                          The number of vertical toothpicks
.                                          equals the number of parts.
.
.                     _                                      _
                    _|1                                    _|1
                  _|2 _                                  _|2 _
                _|3  |1                                _|3  |1
              _|4   _|2                              _|4   _|2
            _|5    |2 _                            _|5    |2 _
          _|6     _|3|1                          _|6     _|3|1
        _|7      |3  |2                        _|7      |3  |2
      _|8       _|4 _|3                      _|8       _|4 _|3
     |9        |4  |2                       |9        |4  |2
               |5  |3
                   |4
.
         Figure 5.                                Figure 6.
.
Then we join the sub-diagrams              Finally we erase the parts that
forming staircases (or zig-zag             are beyond a certain level (in
paths) that represent the                  this case beyond the 9th level)
partitions that have the same              to make the diagram more standard.
number of parts.
.
The numbers in the k-th staircase (from left to right) are the elements of the k-th column of the triangular array.
Note that this diagram is essentially the same diagram used to represent the triangles A237048, A235791, A237591, and other related sequences such as A001227, A060831 and A204217.
There is an infinite family of this kind of triangles, which are related to polygonal numbers and partitions into consecutive parts that differ by d. For more information see the theorems in A285914 and A303300.
Note that if we take two images of the diagram mirroring each other, with the y-axis in the middle of them, then a new diagram is formed, which is symmetric and represents the sequence A237593 as an isosceles triangle. Then if we fold each level (or row) of that isosceles triangle we essentially obtain the structure of the pyramid described in A245092 whose terraces at the n-th level have a total area equal to sigma(n) = A000203(n). (End)

Crossrefs

Another version of A286000.
Tables of the same family where the consecutive parts differ by d are A010766 (d=0), this sequence (d=1), A332266 (d=2), A334945 (d=3), A334618(d=4).
Cf. A000217, A001227, A003056, A109814, A196020, A204217, A211343, A235791, A236104, A237048, A237591, A237593, A245092, A262626, A280850, A280851, A285914, A286013, A288529, A288772, A288773, A288774, A296508, A299765, A303300.

A286000 A table of partitions into consecutive parts (see Comments lines for definition).

Original entry on oeis.org

1, 2, 3, 2, 4, 1, 5, 3, 6, 2, 3, 7, 4, 2, 8, 3, 1, 9, 5, 4, 10, 4, 3, 4, 11, 6, 2, 3, 12, 5, 5, 2, 13, 7, 4, 1, 14, 6, 3, 5, 15, 8, 6, 4, 5, 16, 7, 5, 3, 4, 17, 9, 4, 2, 3, 18, 8, 7, 6, 2, 19, 10, 6, 5, 1, 20, 9, 5, 4, 6, 21, 11, 8, 3, 5, 6, 22, 10, 7, 7, 4, 5, 23, 12, 6, 6, 3, 4, 24, 11, 9, 5, 2, 3, 25, 13, 8, 4, 7, 2
Offset: 1

Views

Author

Omar E. Pol, Apr 30 2017

Keywords

Comments

This is a triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists successive blocks of k consecutive terms in decreasing order, where the m-th block starts with k + m - 1, m>=1, and the first element of column k is in the row k*(k+1)/2.
The partitions of n into consecutive parts are represented from the row n up to row A288529(n) as maximum, exclusively in the columns where the blocks begin.
More precisely, the partition of n into exactly k consecutive parts (if such partition exists) is represented in the column k from the row n up to row n + k - 1 (see examples).
A288772(n) is the minimum number of rows that are required to represent in this table the partitions of all positive integers <= n into consecutive parts.
A288773(n) is the largest of all positive integers whose partitions into consecutive parts can be totally represented in the first n rows of this table.
A288774(n) is the largest positive integers whose partitions into consecutive parts can be totally represented in the first n rows of this table.
Theorem: the smallest part of the partition of n into exactly k consecutive parts (if such partition exists) equals the number of positive integers <= n having a partition into exactly k consecutive parts.

Examples

			Table de partitions into consecutive parts (first 28 rows):
1;
2;
3,   2;
4,   1;
5,   3;
6,   2,  3;
7,   4,  2;
8,   3,  1;
9,   5,  4;
10,  4,  3,  4;
11,  6,  2,  3;
12,  5,  5,  2;
13,  7,  4,  1;
14,  6,  3,  5;
15,  8,  6,  4,  5;
16,  7,  5,  3,  4;
17,  9,  4,  2,  3;
18,  8,  7,  6,  2;
19, 10,  6,  5,  1;
20,  9,  5,  4,  6;
21, 11,  8,  3,  5,  6;
22, 10,  7,  7,  4,  5;
23, 12,  6,  6,  3,  4;
24, 11,  9,  5,  2,  3;
25, 13,  8,  4,  7,  2;
26, 12,  7,  8,  6,  1;
27, 14, 10,  7,  5,  7;
28, 13,  9,  6,  4,  6,  7;
...
Figures A..G show the location (in the columns of the table) of the partitions of n = 1..7 (respectively) into consecutive parts:
.   ------------------------------------------------------------------------
Fig:   A     B       C         D          E            F             G
.   ------------------------------------------------------------------------
. n:   1     2       3         4          5            6             7
Row ------------------------------------------------------------------------
1   | [1];|  1; |  1;     |  1;    |  1;        |  1;         |  1;        |
2   |     | [2];|  2;     |  2;    |  2;        |  2;         |  2;        |
3   |     |     | [3],[2];|  3;  2;|  3,  2;    |  3,  2;     |  3,  2;    |
4   |     |     |  4 ,[1];| [4], 1;|  4,  1;    |  4,  1;     |  4,  1;    |
5   |     |     |         |        | [5],[3];   |  5,  3;     |  5,  3;    |
6   |     |     |         |        |  6, [2], 3;| [6], 2, [3];|  6,  2,  3;|
7   |     |     |         |        |            |  7,  4, [2];| [7],[4], 2;|
8   |     |     |         |        |            |  8,  3, [1];|  8, [3], 1;|
.   ------------------------------------------------------------------------
Figure F: for n = 6 the partitions of 6 into consecutive parts are [6] and [3, 2, 1]. These partitions have 1 and 3 consecutive parts respectively. On the other hand  we can find the mentioned partitions in the columns 1 and 3 of this table, starting at the row 6.
.
Figures H..K show the location (in the columns of the table) of the partitions of 8..11 (respectively) into consecutive parts:
.    --------------------------------------------------------------------
Fig:        H             I                  J                 K
.    --------------------------------------------------------------------
. n:        8             9                  10                11
Row  --------------------------------------------------------------------
1    |  1;        |  1;            |   1;             |   1;            |
1    |  2;        |  2;            |   2;             |   2;            |
3    |  3,  2;    |  3,  2;        |   3,  2;         |   3,  2;        |
4    |  4,  1;    |  4,  1;        |   4,  1;         |   4,  1;        |
5    |  5,  3;    |  5,  3;        |   5,  3;         |   5,  3;        |
6    |  6,  2,  3;|  6,  2,  3;    |   6,  2,  3;     |   6,  2,  3;    |
7    |  7,  4,  2;|  7,  4,  2;    |   7,  4,  2;     |   7,  4,  2;    |
8    | [8], 3,  1;|  8,  3,  1;    |   8,  3,  1;     |   8,  3,  1;    |
9    |            | [9],[5],[4];   |   9,  5,  4;     |   9,  5,  4;    |
10   |            | 10, [4],[3], 4;| [10], 4,  3, [4];|  10,  4,  3;  4;|
11   |            | 11,  6, [2], 3;|  11,  6,  2; [3];| [11],[6], 2,  3;|
12   |            |                |  12,  5,  5, [2];|  12, [5], 5,  2;|
13   |            |                |  13,  7,  4, [1];|  13,  7,  4,  1;|
.    --------------------------------------------------------------------
Figure J: For n = 10 the partitions of 10 into consecutive parts are [10] and [4, 3, 2, 1]. These partitions have 1 and 4 consecutive parts respectively. On the other hand  we can find the mentioned partitions in the columns 1 and 4 of this table, starting at the row 10.
Illustration of initial terms arranged into the diagram of the triangle A237591:
.                                                           _
.                                                         _|1|
.                                                       _|2 _|
.                                                     _|3  |2|
.                                                   _|4   _|1|
.                                                 _|5    |3 _|
.                                               _|6     _|2|3|
.                                             _|7      |4  |2|
.                                           _|8       _|3 _|1|
.                                         _|9        |5  |4 _|
.                                       _|10        _|4  |3|4|
.                                     _|11         |6   _|2|3|
.                                   _|12          _|5  |5  |2|
.                                 _|13           |7    |4 _|1|
.                               _|14            _|6   _|3|5 _|
.                             _|15             |8    |6  |4|5|
.                           _|16              _|7    |5  |3|4|
.                         _|17               |9     _|4 _|2|3|
.                       _|18                _|8    |7  |6  |2|
.                     _|19                 |10     |6  |5 _|1|
.                   _|20                  _|9     _|5  |4|6 _|
.                 _|21                   |11     |8   _|3|5|6|
.               _|22                    _|10     |7  |7  |4|5|
.             _|23                     |12      _|6  |6  |3|4|
.           _|24                      _|11     |9    |5 _|2|3|
.         _|25                       |13       |8   _|4|7  |2|
.       _|26                        _|12      _|7  |8  |6 _|1|
.     _|27                         |14       |10   |7  |5|7 _|
.    |28                           |13       |9    |6  |4|6|7|
...
The number of horizontal line segments in the n-th row of the diagram equals A001227(n), the number of partitions of n into consecutive parts.

Crossrefs

Row n has length A003056(n).
The first element of column k is in row A000217(k).
For another version see A286001.
Cf. A001227, A109814, A196020, A204217, A235791, A236104, A237048, A237591, A237593, A245092, A285914, A286013, A288529, A288772, A288773, A288774.

A299765 Irregular triangle read by rows, T(n,k), n >= 1, k >= 1, in which row n lists the partitions of n into consecutive parts, with the partitions ordered by increasing number of parts.

Original entry on oeis.org

1, 2, 3, 2, 1, 4, 5, 3, 2, 6, 3, 2, 1, 7, 4, 3, 8, 9, 5, 4, 4, 3, 2, 10, 4, 3, 2, 1, 11, 6, 5, 12, 5, 4, 3, 13, 7, 6, 14, 5, 4, 3, 2, 15, 8, 7, 6, 5, 4, 5, 4, 3, 2, 1, 16, 17, 9, 8, 18, 7, 6, 5, 6, 5, 4, 3, 19, 10, 9, 20, 6, 5, 4, 3, 2, 21, 11, 10, 8, 7, 6, 6, 5, 4, 3, 2, 1, 22, 7, 6, 5, 4, 23, 12, 11
Offset: 1

Views

Author

Omar E. Pol, Feb 26 2018

Keywords

Comments

In the triangle the first partition with m parts appears as the last partition in row A000217(m), m >= 1. - Omar E. Pol, Mar 23 2022
For m >= 0, row 2^m consists of just one element (2^m). - Paolo Xausa, May 24 2025

Examples

			Triangle begins:
   [1];
   [2];
   [3], [2, 1];
   [4];
   [5], [3, 2];
   [6], [3, 2, 1];
   [7], [4, 3];
   [8];
   [9], [5, 4], [4, 3, 2];
  [10], [4, 3, 2, 1];
  [11], [6, 5];
  [12], [5, 4, 3];
  [13], [7, 6];
  [14], [5, 4, 3, 2];
  [15], [8, 7], [6, 5, 4], [5, 4, 3, 2, 1];
  [16];
  [17], [9, 8];
  [18], [7, 6, 5], [6, 5, 4, 3];
  [19], [10, 9];
  [20], [6, 5, 4, 3, 2];
  [21], [11, 10], [8, 7, 6], [6, 5, 4, 3, 2, 1];
  [22], [7, 6, 5, 4];
  [23], [12, 11];
  [24], [9, 8, 7];
  [25], [13, 12], [7, 6, 5, 4, 3];
  [26], [8, 7, 6, 5];
  [27], [14, 13], [10, 9, 8], [7, 6, 5, 4, 3, 2];
  [28], [7, 6, 5, 4, 3, 2, 1];
...
Note that in the below diagram the number of horizontal line segments in the n-th row equals A001227(n), the number of partitions of n into consecutive parts, so we can find the partitions of n into consecutive parts as follows: consider the vertical blocks of numbers that start exactly in the n-th row of the diagram, for example: for n = 15 consider the vertical blocks of numbers that start exactly in the 15th row. They are [15], [8, 7], [6, 5, 4] and [5, 4, 3, 2, 1], equaling the 15th row of the above triangle.
.                                                           _
.                                                         _|1|
.                                                       _|2 _|
.                                                     _|3  |2|
.                                                   _|4   _|1|
.                                                 _|5    |3 _|
.                                               _|6     _|2|3|
.                                             _|7      |4  |2|
.                                           _|8       _|3 _|1|
.                                         _|9        |5  |4 _|
.                                       _|10        _|4  |3|4|
.                                     _|11         |6   _|2|3|
.                                   _|12          _|5  |5  |2|
.                                 _|13           |7    |4 _|1|
.                               _|14            _|6   _|3|5 _|
.                             _|15             |8    |6  |4|5|
.                           _|16              _|7    |5  |3|4|
.                         _|17               |9     _|4 _|2|3|
.                       _|18                _|8    |7  |6  |2|
.                     _|19                 |10     |6  |5 _|1|
.                   _|20                  _|9     _|5  |4|6 _|
.                 _|21                   |11     |8   _|3|5|6|
.               _|22                    _|10     |7  |7  |4|5|
.             _|23                     |12      _|6  |6  |3|4|
.           _|24                      _|11     |9    |5 _|2|3|
.         _|25                       |13       |8   _|4|7  |2|
.       _|26                        _|12      _|7  |8  |6 _|1|
.     _|27                         |14       |10   |7  |5|7 _|
.    |28                           |13       |9    |6  |4|6|7|
...
The diagram is infinite. For more information about the diagram see A286000.
For an amazing connection with sum of divisors function (A000203) see A237593.

Links

Crossrefs

Row n has length A204217(n).
Row sums give A245579.
Right border gives A118235.
Column 1 gives A000027.
Records give A000027.
The number of partitions into consecutive parts in row n is A001227(n).
For tables of partitions into consecutive parts see A286000 and A286001.
Cf. A328365 (mirror).
Cf. A352425 (a subsequence).
Cf. A000203, A000217, A026792, A235791, A237048, A237591, A237593, A245092, A285914, A286013, A288529, A288772, A288773, A288774.

Programs

  • Mathematica
    intervals[n_]:=Module[{x,y},SolveValues[(x^2-y^2+x+y)/2==n&&0A299765row[n_]:=Flatten[SortBy[Map[Range[First[#],Last[#],-1]&,intervals[n]],Length]];
nrows=25;Array[A299765row,nrows] (* Paolo Xausa, Jun 19 2022 *)
  • PARI
    iscons(p) = my(v = vector(#p-1, k, p[k+1] - p[k])); v == vector(#p-1, i, 1);
row(n) = my(list = List()); forpart(p=n, if (iscons(p), listput(list, Vecrev(p)));); Vec(list); \\ Michel Marcus, May 11 2022

Extensions

Name clarified by Omar E. Pol, May 11 2022

A328365 Irregular triangle read by rows, T(n,k), n >= 1, k >= 1, in which row n lists in reverse order the partitions of n into consecutive parts.

Original entry on oeis.org

1, 2, 1, 2, 3, 4, 2, 3, 5, 1, 2, 3, 6, 3, 4, 7, 8, 2, 3, 4, 4, 5, 9, 1, 2, 3, 4, 10, 5, 6, 11, 3, 4, 5, 12, 6, 7, 13, 2, 3, 4, 5, 14, 1, 2, 3, 4, 5, 4, 5, 6, 7, 8, 15, 16, 8, 9, 17, 3, 4, 5, 6, 5, 6, 7, 18, 9, 10, 19, 2, 3, 4, 5, 6, 20, 1, 2, 3, 4, 5, 6, 6, 7, 8, 10, 11, 21, 4, 5, 6, 7, 22, 11, 12, 23, 7, 8, 9, 24
Offset: 1

Views

Author

Omar E. Pol, Oct 22 2019

Keywords

Comments

For m >= 0, row 2^m consists of just one element (2^m). - Paolo Xausa, May 24 2025

Examples

			Triangle begins:
  [1];
  [2];
  [1, 2], [3];
  [4];
  [2, 3], [5];
  [1, 2, 3], [6];
  [3, 4], [7];
  [8];
  [2, 3, 4], [4, 5], [9];
  [1, 2, 3, 4], [10];
  [5, 6], [11];
  [3, 4, 5], [12];
  [6, 7], [13];
  [2, 3, 4, 5], [14];
  [1, 2, 3, 4, 5], [4, 5, 6], [7, 8], [15];
  [16];
  [8, 9], [17];
  [3, 4, 5, 6], [5, 6, 7], [18];
  [9, 10], [19];
  [2, 3, 4, 5, 6], [20];
  [1, 2, 3, 4, 5, 6], [6, 7, 8], [10, 11], [21];
  [4, 5, 6, 7], [22];
  [11, 12], [23];
  [7, 8, 9], [24];
  [3, 4, 5, 6, 7], [12, 13], [25];
  [5, 6, 7, 8], [26];
  [2, 3, 4, 5, 6, 7], [8, 9, 10], [13, 14], [27];
  [1, 2, 3, 4, 5, 6, 7], [28];
  ...
For n = 9 there are three partitions of 9 into consecutive parts, they are [9], [5, 4], [4, 3, 2], so the 9th row of triangle is [2, 3, 4], [4, 5], [9].
Note that in the below diagram the number of horizontal line segments in the n-th row equals A001227(n), the number of partitions of n into consecutive parts, so we can find the partitions of n into consecutive parts as follows: consider the vertical blocks of numbers that start exactly in the n-th row of the diagram, for example: for n = 15 consider the vertical blocks of numbers that start exactly in the 15th row. They are [1, 2, 3, 4, 5], [4, 5, 6], [7, 8], [15], equaling the 15th row of the above triangle.
Row        _
  1       |1|_
  2       |_ 2|_
  3       |1|  3|_
  4       |2|_   4|_
  5       |_ 2|    5|_
  6       |1|3|_     6|_
  7       |2|  3|      7|_
  8       |3|_ 4|_       8|_
  9       |_ 2|  4|        9|_
  10      |1|3|  5|_        10|_
  11      |2|4|_   5|         11|_
  12      |3|  3|  6|_          12|_
  13      |4|_ 4|    6|           13|_
  14      |_ 2|5|_   7|_            14|_
  15      |1|3|  4|    7|             15|_
  16      |2|4|  5|    8|_              16|_
  17      |3|5|_ 6|_     8|               17|_
  18      |4|  3|  5|    9|_                18|_
  19      |5|_ 4|  6|      9|                 19|_
  20      |_ 2|5|  7|_    10|_                  20|_
  21      |1|3|6|_   6|     10|                   21|_
  22      |2|4|  4|  7|     11|_                    22|_
  23      |3|5|  5|  8|_      11|                     23|_
  24      |4|6|_ 6|    7|     12|_                      24|_
  25      |5|  3|7|_   8|       12|                       25|_
  26      |6|_ 4|  5|  9|_      13|_                        26|_
  27      |_ 2|5|  6|    8|       13|                         27|_
  28      |1|3|6|  7|    9|       14|                           28|
  ...
The diagram is infinite. For more information about the diagram see A286001.
For an amazing connection with sum of divisors function (A000203) see A237593.

Links

Crossrefs

Mirror of A299765.
Row n has length A204217(n).
Row sums give A245579.
Column 1 gives A118235.
Right border gives A000027.
Records give A000027.
Where records occur gives A285899.
The number of partitions into consecutive parts in row n is A001227(n).
For tables of partitions into consecutive parts see A286000 and A286001.
Cf. A000203, A026792, A235791, A237048, A237591, A237593, A245092, A285914, A286013, A288529, A288772, A288773, A288774, A328361, A328362.

Programs

  • Mathematica
    Table[With[{h = Floor[n/2] - Boole[EvenQ@ n]},Append[Array[Which[Total@ # == n, #, Total@ Most@ # == n, Most[#], True, Nothing] &@ NestWhile[Append[#, #[[-1]] + 1] &, {#}, Total@ # <= n &, 1, h - # + 1] &, h], {n}]], {n, 24}] // Flatten (* Michael De Vlieger, Oct 22 2019 *)

A288529 a(n) is the minimum number of rows from the table described in A286000 that are required to represent the partitions of n into consecutive parts.

Original entry on oeis.org

1, 2, 4, 4, 6, 8, 8, 8, 11, 13, 12, 14, 14, 17, 19, 16, 18, 21, 20, 24, 26, 25, 24, 26, 29, 29, 32, 34, 30, 34, 32, 32, 38, 37, 41, 43, 38, 41, 44, 44, 42, 48, 44, 51, 53, 49, 48, 50, 55, 54, 56, 59, 54, 62, 64, 62, 62, 61, 60, 67, 62, 65, 71, 64, 74, 76, 68, 75, 74, 76, 72, 80, 74, 77, 84, 83, 87, 89, 80, 84, 89, 85
Offset: 1

Views

Author

Omar E. Pol, Jun 19 2017

Keywords

Comments

a(n) has the same definition related to the table A286001 which is another version of the table A286000.
First differs from A288772 at a(11), which shares infinitely many terms.

Examples

			Figures A..D show the evolution of the table of partitions into consecutive parts described in A286000, for n = 8..11:
.     ---------------------------------------------------------------------
Figure:      A            B                    C                  D
.     ---------------------------------------------------------------------
.    n:      8            9                   10                 11
Row   ---------------------------------------------------------------------
1     |  1;        |  1;             |   1;             |   1;            |
1     |  2;        |  2;             |   2;             |   2;            |
3     |  3,  2;    |  3,  2;         |   3,  2;         |   3,  2;        |
4     |  4,  1;    |  4,  1;         |   4,  1;         |   4,  1;        |
5     |  5,  3;    |  5,  3;         |   5,  3;         |   5,  3;        |
6     |  6,  2,  3;|  6,  2,  3;     |   6,  2,  3;     |   6,  2,  3;    |
7     |  7,  4,  2;|  7,  4,  2;     |   7,  4,  2;     |   7,  4,  2;    |
8     | [8], 3,  1;|  8,  3,  1;     |   8,  3,  1;     |   8,  3,  1;    |
9     |            | [9],[5],[4];    |   9,  5,  4;     |   9,  5,  4;    |
10    |            | 10, [4],[3],  4;| [10], 4,  3, [4];|  10,  4,  3;  4;|
11    |            | 11,  6, [2],  3;|  11,  6,  2; [3];| [11],[6], 2,  3;|
12    |            |                 |  12,  5,  5, [2];|  12, [5], 5,  2;|
13    |            |                 |  13,  7,  4, [1];|                 |
.     ---------------------------------------------------------------------
. a(n):      8              11                13                 12
.     ---------------------------------------------------------------------
For n = 8 we need a table with at least 8 rows, so a(8) = 8.
For n = 9 we need a table with at least 11 rows, so a(9) = 11.
For n = 10 we need a table with at least 13 rows, so a(10) = 13.
For n = 11 we need a table with at least 12 rows, so a(11) = 12.

Crossrefs

Cf. A109814, A237593, A286000, A286001, A288772, A288773, A288774.

Formula

a(n) = A109814(n) + n - 1.

A288772 a(n) is the minimum number of rows from the table described in A286000 that are required to represent the partitions of all positive integers <= n into consecutive parts.

Original entry on oeis.org

1, 2, 4, 4, 6, 8, 8, 8, 11, 13, 13, 14, 14, 17, 19, 19, 19, 21, 21, 24, 26, 26, 26, 26, 29, 29, 32, 34, 34, 34, 34, 34, 38, 38, 41, 43, 43, 43, 44, 44, 44, 48, 48, 51, 53, 53, 53, 53, 55, 55, 56, 59, 59, 62, 64, 64, 64, 64, 64, 67, 67, 67, 71, 71, 74, 76, 76, 76, 76, 76, 76, 80, 80, 80, 84, 84, 87, 89, 89, 89, 89
Offset: 1

Views

Author

Omar E. Pol, Jun 17 2017

Keywords

Comments

a(n) has the same definition related to the table A286001 which is another version of the table A286000.
First differs from A288529 at a(11), which shares infinitely many terms.

Examples

			Figures A..D show the evolution of the table of partitions into consecutive parts described in A286000, for n = 8..11:
.     ---------------------------------------------------------------------
Figure:      A            B                    C                  D
.     ---------------------------------------------------------------------
.    n:      8            9                   10                 11
Row   ---------------------------------------------------------------------
1     |  1;        |  1;             |   1;             |   1;            |
1     |  2;        |  2;             |   2;             |   2;            |
3     |  3,  2;    |  3,  2;         |   3,  2;         |   3,  2;        |
4     |  4,  1;    |  4,  1;         |   4,  1;         |   4,  1;        |
5     |  5,  3;    |  5,  3;         |   5,  3;         |   5,  3;        |
6     |  6,  2,  3;|  6,  2,  3;     |   6,  2,  3;     |   6,  2,  3;    |
7     |  7,  4,  2;|  7,  4,  2;     |   7,  4,  2;     |   7,  4,  2;    |
8     | [8], 3,  1;|  8,  3,  1;     |   8,  3,  1;     |   8,  3,  1;    |
9     |            | [9],[5],[4];    |   9,  5,  4;     |   9,  5,  4;    |
10    |            | 10, [4],[3],  4;| [10], 4,  3, [4];|  10,  4,  3;  4;|
11    |            | 11,  6, [2],  3;|  11,  6,  2; [3];| [11],[6], 2,  3;|
12    |            |                 |  12,  5,  5, [2];|  12, [5], 5,  2;|
13    |            |                 |  13,  7,  4, [1];|  13,  7,  4,  1;|
.     ---------------------------------------------------------------------
. a(n):      8              11                13                 13
.     ---------------------------------------------------------------------
For n = 8 we need a table with at least 8 rows, so a(8) = 8.
For n = 9 we need a table with at least 11 rows, so a(9) = 11.
For n = 10 we need a table with at least 13 rows, so a(10) = 13.
For n = 11 we need a table with at least 13 rows, so a(11) = 13.

Crossrefs

Cf. A001227, A109814, A204217, A237593, A286000, A286001, A288529, A288773, A288774.

A288773 a(n) is the largest of all positive integers whose partitions into consecutive parts can be totally represented in the first n rows of the table described in A286000.

Original entry on oeis.org

1, 2, 2, 4, 4, 5, 5, 8, 8, 8, 9, 9, 11
Offset: 1

Views

Author

Omar E. Pol, Jun 17 2017

Keywords

Comments

a(n) has the same definition related to the table A286001 which is another version of the table A286000.
First differs from A288774 at a(12), which shares infinitely many terms.

Examples

			Figures A, B, C show the evolution of the table of partitions into consecutive parts described in A286000, with 11, 12 and 13 rows respectively:
.     ------------------------------------------------------
Figure:       A                B                  C
------------------------------------------------------------
.   n =      11               12                 13
Row   ------------------------------------------------------
1     |  1;            |   1;            |   1;            |
1     |  2;            |   2;            |   2;            |
3     |  3,  2;        |   3,  2;        |   3,  2;        |
4     |  4,  1;        |   4,  1;        |   4,  1;        |
5     |  5,  3;        |   5,  3;        |   5,  3;        |
6     |  6,  2,  3;    |   6,  2,  3;    |   6,  2,  3;    |
7     |  7,  4,  2;    |   7,  4,  2;    |   7,  4,  2;    |
8     |  8,  3,  1;    |   8,  3,  1;    |   8,  3,  1;    |
9     | [9],[5],[4];   |  [9],[5],[4];   |   9,  5,  4;    |
10    | 10, [4],[3], 4;|  10, [4],[3], 4;|  10,  4,  3;  4;|
11    | 11,  6, [2], 3;|  11,  6, [2]; 3;| [11],[6], 2,  3;|
12    |                |  12,  5,  5,  2;|  12, [5], 5,  2;|
13    |                |                 |  13,  7,  4,  1;|
.     ------------------------------------------------------
. a(n):       9                 9                 11
.     ------------------------------------------------------
For n = 11, in the first 11 rows of the table can be represented the partitions into consecutive parts of the integers 1, 2, 3, 4, 5, 6, 7, 8 and 9. The largest of these positive integers is 9, so a(11) = 9.
For n = 12, in the first 12 rows of the table can be represented the partitions into consecutive parts of the integers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 11. The largest of these positive integers is 11, but the partitions into consecutive parts of 10 cannot be represented, so a(12) = 9, not 11.
For n = 13, in the first 13 rows of the table can be represented the partitions into consecutive parts of the integers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11. The largest of these positive integers is 11, so a(13) = 11.

Crossrefs

Cf. A237593, A286000, A286001, A288529, A288772, A288774.

A319895 a(n) is the number of partitions of n into consecutive parts, plus the total number of parts in those partitions.

Original entry on oeis.org

2, 2, 5, 2, 5, 6, 5, 2, 9, 7, 5, 6, 5, 7, 15, 2, 5, 11, 5, 8, 16, 7, 5, 6, 11, 7, 16, 10, 5, 17, 5, 2, 16, 7, 19, 15, 5, 7, 16, 8, 5, 19, 5, 11, 32, 7, 5, 6, 13, 13, 16, 11, 5, 21, 22, 10, 16, 7, 5, 21, 5, 7, 34, 2, 22, 23, 5, 11, 16, 21, 5, 16, 5, 7, 33, 11, 25, 24, 5, 8, 26, 7, 5, 23, 22, 7, 16, 14, 5
Offset: 1

Views

Author

Omar E. Pol, Sep 30 2018

Keywords

Comments

a(n) is also the total length of all pairs of orthogonal line segments whose horizontal and upper parts are in the n-th row of the diagram associated to partitions into consecutive parts as shown in the Example section.
a(n) = 2 iff n is a power of 2.
a(n) = 5 iff n is an odd prime.

Examples

			Illustration of a diagram of partitions into consecutive parts (first 28 rows):
.                                                           _
.                                                         _|1
.                                                       _|2 _
.                                                     _|3  |2
.                                                   _|4   _|1
.                                                 _|5    |3 _
.                                               _|6     _|2|3
.                                             _|7      |4  |2
.                                           _|8       _|3 _|1
.                                         _|9        |5  |4 _
.                                       _|10        _|4  |3|4
.                                     _|11         |6   _|2|3
.                                   _|12          _|5  |5  |2
.                                 _|13           |7    |4 _|1
.                               _|14            _|6   _|3|5 _
.                             _|15             |8    |6  |4|5
.                           _|16              _|7    |5  |3|4
.                         _|17               |9     _|4 _|2|3
.                       _|18                _|8    |7  |6  |2
.                     _|19                 |10     |6  |5 _|1
.                   _|20                  _|9     _|5  |4|6 _
.                 _|21                   |11     |8   _|3|5|6
.               _|22                    _|10     |7  |7  |4|5
.             _|23                     |12      _|6  |6  |3|4
.           _|24                      _|11     |9    |5 _|2|3
.         _|25                       |13       |8   _|4|7  |2
.       _|26                        _|12      _|7  |8  |6 _|1
.     _|27                         |14       |10   |7  |5|7 _
.    |28                           |13       |9    |6  |4|6|7
...
For n = 21 we have that there are four partitions of 21 into consecutive parts, they are [21], [11, 10], [8, 7, 6], [6, 5, 4, 3, 2, 1]. The total number of parts is 1 + 2 + 3 + 6 = 12. Therefore the number of partitions plus the total number of parts is 4 + 12 = 16, so a(21) = 16.
On the other hand, in the above diagram there are four pairs of orthogonal line segments whose horizontal upper part are located on the 21st row, as shown below:
.                   _                     _       _         _
.                  |21                   |11     |8        |6
.                                        |10     |7        |5
.                                                |6        |4
.                                                          |3
.                                                          |2
.                                                          |1
.
The four horizontal line segments have length 1, and the vertical line segments have lengths 1, 2, 3, 6 respectively. Therefore the total length of the line segments is 1 + 1 + 1 + 1 + 1 + 2 + 3 + 6 = 16, so a(21) = 16.

Links

Crossrefs

For tables of partitions into consecutive parts see A286000 and A286001.
Cf. A000079, A001227, A065091, A204217, A237048, A237593, A285898, A285899, A285900, A285900, A285901, A285902, A288529, A288772, A288773, A288774, A299765.

Programs

Formula

a(n) = A001227(n) + A204217(n).

Extensions

Term a(87) corrected from 6 to 16 by Antti Karttunen, Dec 06 2021
Showing 1-8 of 8 results.

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