A379631 -id:A379631 - cp's OEIS frontend

A379634 Irregular triangle read by rows in which row n lists the odd divisors of n ordered as the mirror of A261697.

1, 1, 3, 1, 1, 5, 1, 3, 1, 7, 1, 1, 3, 9, 1, 5, 1, 11, 1, 3, 1, 13, 1, 7, 1, 5, 3, 15, 1, 1, 17, 1, 9, 3, 1, 19, 1, 5, 1, 7, 3, 21, 1, 11, 1, 23, 1, 3, 1, 5, 25, 1, 13, 1, 9, 3, 27, 1, 7, 1, 29, 1, 5, 15, 3, 1, 31, 1, 1, 11, 3, 33, 1, 17, 1, 7, 5, 35, 1, 9, 3, 1, 37, 1, 19, 1, 13, 3, 39, 1, 5, 1, 41, 1, 7, 21, 3, 1, 43, 1, 11, 1

Offset: 1

Omar E. Pol, Dec 31 2024

Row n gives the last A001227(n) terms of the n-th row of A379630 and of A379631.

For a correspondence between the row n and the partitions of n into consecutive parts see A379630.

			Triangle begins:
   1;
   1;
   3,  1;
   1;
   5,  1;
   3,  1;
   7,  1;
   1;
   3,  9,  1;
   5,  1;
  11,  1;
   3,  1;
  13,  1;
   7,  1;
   5,  3, 15,  1;
   1;
  17,  1;
   9,  3,  1;
  19,  1;
   5,  1;
   7,  3, 21,  1;
  11,  1;
  23,  1;
   3,  1;
   5, 25,  1;
  13,  1;
   9,  3, 27,  1;
   7,  1;
  ...
Illustration of initial terms:
   Row    _
   1     |1|_
   2     |_ 1|_
   3     |3|  1|_
   4     | |_   1|_
   5     |_ 5|    1|_
   6     |3| |_     1|_
   7     | |  7|      1|_
   8     | |_  |_       1|_
   9     |_ 3|  9|        1|_
  10     |5| |   |_         1|_
  11     | | |_  11|          1|_
  12     | |  3|   |_           1|_
  13     | |_  |   13|            1|_
  14     |_ 7| |_    |_             1|_
  15     |5| |  3|   15|              1|_
  16     | | |   |     |_               1|_
  17     | | |_  |_    17|                1|_
  18     | |  9|  3|     |_                 1|_
  19     | |_  |   |     19|                  1|_
  20     |_ 5| |   |_      |_                   1|_
  21     |7| | |_   3|     21|                    1|_
  22     | | | 11|   |       |_                     1|_
  23     | | |   |   |_      23|                      1|_
  24     | | |_  |    3|       |_                       1|_
  25     | |  5| |_    |       25|                        1|_
  26     | |_  | 13|   |_        |_                          1|_
  27     |_ 9| |   |    3|       27|                           1|_
  28     |7| | |   |     |         |                             1|
  ...
The diagram is also the right part of the diagram of A379630 and of A379631.
The geometrical structure is the same as the diagram of A261350 which is the mirror of A237591.

Mirror of A261697.
Right border gives A000012.
Row lengths give A001227.
Row sums give A000593.
Other versions are A182469, A261697, A261698.
Cf. A196020, A235791, A236104, A237048, A237591, A237593, A261350, A261699, A379630, A379631, A379632, A379633.

A379632 Irregular triangle read by rows in which row n lists the smallest parts of the partitions of n into consecutive parts (with the partitions ordered by increasing number of parts).

Original entry on oeis.org

1, 2, 3, 1, 4, 5, 2, 6, 1, 7, 3, 8, 9, 4, 2, 10, 1, 11, 5, 12, 3, 13, 6, 14, 2, 15, 7, 4, 1, 16, 17, 8, 18, 5, 3, 19, 9, 20, 2, 21, 10, 6, 1, 22, 4, 23, 11, 24, 7, 25, 12, 3, 26, 5, 27, 13, 8, 2, 28, 1, 29, 14, 30, 9, 6, 4, 31, 15, 32, 33, 16, 10, 3, 34, 7, 35, 17, 5, 2, 36, 11, 1, 37, 18, 38, 8, 39, 19, 12, 4, 40, 6, 41, 20

Offset: 1

Omar E. Pol, Dec 31 2024

Row n gives the first A001227(n) terms of the n-th row of A379630.

			Triangle begins:
   1;
   2;
   3,  1;
   4;
   5,  2;
   6,  1;
   7,  3;
   8;
   9,  4,  2;
  10,  1;
  11,  5;
  12,  3;
  13,  6;
  14,  2;
  15,  7,  4,  1;
  16;
  17,  8;
  18,  5,  3;
  19,  9;
  20,  2;
  21, 10,  6,  1;
  22,  4;
  23, 11;
  24,  7;
  25, 12,  3;
  26,  5;
  27, 13,  8,  2;
  28,  1;
  ...
Illustration of initial terms:
                                                         _
                                                       _|1|
                                                     _|2 _|
                                                   _|3  |1|
                                                 _|4   _| |
                                               _|5    |2 _|
                                             _|6     _| |1|
                                           _|7      |3  | |
                                         _|8       _|  _| |
                                       _|9        |4  |2 _|
                                     _|10        _|   | |1|
                                   _|11         |5   _| | |
                                 _|12          _|   |3  | |
                               _|13           |6    |  _| |
                             _|14            _|    _| |2 _|
                           _|15             |7    |4  | |1|
                         _|16              _|     |   | | |
                       _|17               |8     _|  _| | |
                     _|18                _|     |5  |3  | |
                   _|19                 |9      |   |  _| |
                 _|20                  _|      _|   | |2 _|
               _|21                   |10     |6   _| | |1|
             _|22                    _|       |   |4  | | |
           _|23                     |11      _|   |   | | |
         _|24                      _|       |7    |  _| | |
       _|25                       |12       |    _| |3  | |
     _|26                        _|        _|   |5  |  _| |
   _|27                         |13       |8    |   | |2 _|
  |28                           |         |     |   | | |1|
  ...
The diagram is also the left part of the diagram of A379630.
The geometrical structure is the same as the diagram of A237591.

Positive terms of A211343.
Absolute values of A341971.
Column 1 gives A000027.
Right border gives A118235.
Row lengths give A001227.
Row sums give A286014.
Subsequence of A286001 and of A299765 and of A379630.
For the largest parts see A379633.
Cf. A196020, A235791, A236104, A237048, A237591, A237593, A245092, A262626, A379631, A379634.

A379633 Irregular triangle read by rows in which row n lists the largest parts of 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, 4, 5, 3, 6, 3, 7, 4, 8, 9, 5, 4, 10, 4, 11, 6, 12, 5, 13, 7, 14, 5, 15, 8, 6, 5, 16, 17, 9, 18, 7, 6, 19, 10, 20, 6, 21, 11, 8, 6, 22, 7, 23, 12, 24, 9, 25, 13, 7, 26, 8, 27, 14, 10, 7, 28, 7, 29, 15, 30, 11, 9, 8, 31, 16, 32, 33, 17, 12, 8, 34, 10, 35, 18, 9, 8, 36, 13, 8, 37, 19, 38, 11, 39, 20, 14, 9, 40, 10, 41, 21

Offset: 1

Omar E. Pol, Dec 31 2024

Row n gives the first A001227(n) terms of the n-th row of A379631.

Conjecture: the right border gives A212652.

			Triangle begins:
   1;
   2;
   3,  2;
   4;
   5,  3;
   6,  3;
   7,  4;
   8;
   9,  5,  4;
  10,  4;
  11,  6;
  12,  5;
  13,  7;
  14,  5;
  15,  8,  6,  5;
  16;
  17,  9;
  18,  7,  6;
  19, 10;
  20,  6;
  21, 11,  8,  6;
  22,  7;
  23, 12;
  24,  9;
  25, 13,  7;
  26,  8;
  27, 14, 10,  7;
  28,  7;
  ...
Illustration of initial terms:
                                                         _
                                                       _|1|
                                                     _|2 _|
                                                   _|3  |2|
                                                 _|4   _| |
                                               _|5    |3 _|
                                             _|6     _| |3|
                                           _|7      |4  | |
                                         _|8       _|  _| |
                                       _|9        |5  |4 _|
                                     _|10        _|   | |4|
                                   _|11         |6   _| | |
                                 _|12          _|   |5  | |
                               _|13           |7    |  _| |
                             _|14            _|    _| |5 _|
                           _|15             |8    |6  | |5|
                         _|16              _|     |   | | |
                       _|17               |9     _|  _| | |
                     _|18                _|     |7  |6  | |
                   _|19                 |10     |   |  _| |
                 _|20                  _|      _|   | |6 _|
               _|21                   |11     |8   _| | |6|
             _|22                    _|       |   |7  | | |
           _|23                     |12      _|   |   | | |
         _|24                      _|       |9    |  _| | |
       _|25                       |13       |    _| |7  | |
     _|26                        _|        _|   |8  |  _| |
   _|27                         |14       |10   |   | |7 _|
  |28                           |         |     |   | | |7|
  ...
The diagram is also the left part of the diagram of A379631.
The geometrical structure is the same as the diagram of A237591.

Positive terms of A286013.
Column 1 gives A000027.
Row lengths give A001227.
Row sums give A286015.
Subsequence of A286000 and of A299765 and of A379631.
For the smallest parts see A379632.
Cf. A196020, A212652, A235791, A236104, A237048, A237591, A237593, A245092, A262626, A379630, A379634.

cp's OEIS Frontend

A379634 Irregular triangle read by rows in which row n lists the odd divisors of n ordered as the mirror of A261697.

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A379632 Irregular triangle read by rows in which row n lists the smallest parts of the partitions of n into consecutive parts (with the partitions ordered by increasing number of parts).

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A379633 Irregular triangle read by rows in which row n lists the largest parts of the partitions of n into consecutive parts (with the partitions ordered by increasing number of parts).

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Author

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Examples

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