A379634 -id:A379634 - cp's OEIS frontend

A379288 Irregular triangle read by rows in which row n lists the odd divisors of n excluding odd divisors e for which there exists another divisor j with j < e < 2*j.

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

Offset: 1

Omar E. Pol, Dec 21 2024

The excluded divisors are the odd divisors e listed in A005279.
Conjecture 1: the row lengths are given by A237271 (true for at least the first 10000 terms of A237271)
From Hartmut F. W. Hoft, Jan 09 2025: (Start)
Proof of Conjecture 1:
An entire part of SRS(n), n = 2^k * q with k >= 0 and q odd, up to the diagonal is described in row n of A249223 by a 1 in position d, an odd divisor of n, 0's in positions d-1 and 2^(k+1) * f, f >= d an odd divisor of n, and nonzero numbers that increase or decrease by 1 in between.
The odd divisors e of n with d < e < 2^(k+1) * f are the "e" odd divisors of A005279 since for divisor s of n, d < s < e < 2*s < 2^(k+1) * f holds.
The odd divisors u of n greater than A003056(n) are encoded by the 2^(k+1) * f above as u = q/f and odd divisors d < A003056(n) are also encoded as 2^(k+1) * q/d. Then odd divisors e of n with q/f < e < 2^(k+1) * q/d are the "e" odd divisors of A005279 since for divisor t of n,  q/f < t < e < 2*t < 2^(k+1) * q/d holds.
For a part containing the diagonal the inequalities above hold on the respective sides of the diagonal.
As a consequence the number of entries in row n of this triangle equals A237271(n). (End)
From Omar E. Pol, Jun 26 2025: (Start)
Conjecture 2: T(n,m) is the smallest number in the m-th 2-dense sublist of divisors of n.
We call "2-dense sublists of divisors of n" to the maximal sublists of divisors of n whose terms increase by a factor of at most 2.
In a 2-dense sublist of divisors of n the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of n.
If the conjecture is true so row sums give A379379 and the row lengths give A237271, and the same row lengths have the sequences A384222, A384225 and A384226. Also the conjecture of A384149 should be true.
Observation: at least for the first 5000 rows (the first 15542 terms) the conjecture 2  coincides with the definition from the Name section and the row lengths give A237271.
An example of the conjecture 2, for n = 1..24 is as shown below:
  -------------------------------------------------------------------
  |  n |  Row n of        |  List of divisors of n       | Number of |
  |    |  the triangle    |  [with sublists in brackets] | sublists  |
  --------------------------------------------------------------------
  |  1 |   1;             |  [1];                        |     1     |
  |  2 |   1;             |  [1, 2];                     |     1     |
  |  3 |   1,  3;         |  [1], [3];                   |     2     |
  |  4 |   1;             |  [1, 2, 4];                  |     1     |
  |  5 |   1,  5;         |  [1], [5];                   |     2     |
  |  6 |   1;             |  [1, 2, 3, 6];               |     1     |
  |  7 |   1,  7;         |  [1], [7];                   |     2     |
  |  8 |   1;             |  [1, 2, 4, 8];               |     1     |
  |  9 |   1,  3,  9;     |  [1], [3], [9];              |     3     |
  | 10 |   1,  5;         |  [1, 2], [5, 10];            |     2     |
  | 11 |   1, 11;         |  [1], [11];                  |     2     |
  | 12 |   1;             |  [1, 2, 3, 4, 6, 12];        |     1     |
  | 13 |   1, 13;         |  [1], [13];                  |     2     |
  | 14 |   1,  7;         |  [1, 2], [7, 14];            |     2     |
  | 15 |   1,  3, 15;     |  [1], [3, 5], [15];          |     3     |
  | 16 |   1;             |  [1, 2, 4, 8, 16];           |     1     |
  | 17 |   1, 17;         |  [1], [17];                  |     2     |
  | 18 |   1;             |  [1, 2, 3, 6, 9, 18];        |     1     |
  | 19 |   1, 19;         |  [1], [19];                  |     2     |
  | 20 |   1;             |  [1, 2, 4, 5, 10, 20];       |     1     |
  | 21 |   1,  3,  7, 21; |  [1], [3], [7], [21];        |     4     |
  | 22 |   1, 11;         |  [1, 2], [11, 22];           |     2     |
  | 23 |   1, 23;         |  [1], [23];                  |     2     |
  | 24 |   1;             |  [1, 2, 3, 4, 6, 8, 12, 24]; |     1     |
   ...
For n = 10 the list of divisors of 10 is [1, 2, 5, 10]. There are two 2-dense sublists of divisors of 10, they are [1, 2], [5, 10]. The smallest numbers in the sublists are [1, 5] respectively, so the row 10 is [1, 5].
For n = 15 the list of divisors of 15 is [1, 3, 5, 15]. There are three 2-dense sublists of divisors of 15, they are [1], [3, 5], [15]. The smallest numbers in the sublists are [1, 3, 15] respectively, so the row 15 is [1, 3, 15].
78 is the first practical number A005153 not in A174973. For n = 78 the list of divisors of 78 is [1, 2, 3, 6, 13, 26, 39, 78]. There are two 2-dense sublists of divisors of 78, they are [1, 2, 3, 6] and [13, 26, 39, 78]. The smallest numbers in the sublists are [1, 13] respectively, so the row 78 is [1, 13].
(End)
Conjecture 3: T(n,m) is the m-th divisor p of n such that p is greater than twice the adjacent previous divisor of n. - Omar E. Pol, Aug 02 2025

These are the odd terms of A379374.
Subsequence of A182469.
Row sums give A379379.
Cf. A001227, A005279, A237270, A237271, A237593, A239657, A379384, A379461, A379634.
Cf. A000203, A003056, A249223.
Cf. A005153, A174973, A384149, A384222, A384225, A384226, A385000.

row[n_] := Module[{d = Partition[Divisors[n], 2, 1]}, Select[Join[{1}, Select[d, #[[2]] >= 2*#[[1]] &][[;; , 2]]], OddQ]]; Table[row[n], {n, 1, 50}] // Flatten (* Amiram Eldar, Dec 22 2024 *)

More terms from Amiram Eldar, Dec 22 2024

A379631 Irregular triangle read by rows: T(n,m), n >= 1, m >= 1, in which row n lists the largest parts of the partitions of n into consecutive parts followed by the conjugate corresponding odd divisors of n in accordance with the theorem described in A379630.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 30 2024

Consider that the mentioned partitions are ordered by increasing number of parts.

Row n gives the n-th row of A379633 together with the n-th row of A379634.

			Triangle begins:
   1,  1;
   2,  1;
   3,  2,  3,  1;
   4,  1;
   5,  3,  5,  1;
   6,  3,  3,  1;
   7,  4,  7,  1;
   8,  1;
   9,  5,  4,  3,  9,  1,
  10,  4,  5,  1;
  11,  6, 11,  1;
  12,  5,  3,  1;
  13,  7, 13,  1;
  14,  5,  7,  1;
  15,  8,  6,  5,  5,  3, 15,  1;
  16,  1;
  17,  9, 17,  1;
  18,  7,  6,  9,  3,  1;
  19, 10, 19,  1;
  20,  6,  5,  1;
  21, 11,  8,  6,  7,  3, 21,  1;
  ...
For n = 21 the partitions of 21 into consecutive parts are [21], [11, 10], [8, 7, 6], [6, 5, 4, 3, 2, 1].
On the other hand the odd divisors of 21 are [1, 3, 7, 21].
To determine how these partitions are related to the odd divisors we follow the two rules of the theorem described in A379630 as shown below:
The first partition is [21] and the number of parts is 1 and 1 is odd so the corresponding odd divisor of 21 is 1.
The second partition is [11, 10] and the number of parts is 2 and 2 is even so the corresponding odd divisor of 21 is equal to 11 + 10 = 21.
The third partition is [8, 7, 6] and the number of parts is 3 and 3 is odd so the corresponding odd divisor of 21 is 3.
The fourth partition is [6, 5, 4, 3, 2, 1] and the number of parts is 6 and 6 is even so the corresponding odd divisor of 21 is equal to 6 + 1 = 5 + 2 = 4 + 3 = 7.
Summarizing in a table:
  --------------------------------------
              Correspondence
  --------------------------------------
    Partitions of 21              Odd
    into consecutive           divisors
         parts                   of 21
  -------------------         ----------
   [21]   ....................     1
   [11, 10]   ................    21
   [8, 7, 6]  ................     3
   [6, 5, 4, 3, 2, 1]  .......     7
.
Then we can make a table of conjugate correspondence in which the four partitions are arrenged in four columns with the largest parts at the top as shown below:
  ------------------------------------------
           Conjugate correspondence
  ------------------------------------------
    Partitions of 21              Odd
    into consecutive           divisors
    parts as columns             of 21
  -------------------     ------------------
   21   11    8    6       7    3   21    1
    |   10    7    5       |    |    |    |
    |    |    6    4       |    |    |    |
    |    |    |    3       |    |    |    |
    |    |    |    2       |    |    |    |
    |    |    |    1       |    |    |    |
    |    |    |    |_______|    |    |    |
    |    |    |_________________|    |    |
    |    |___________________________|    |
    |_____________________________________|
.
Then removing all rows except the first row we have a table of conjugate correspondence for largest parts and odd divisors as shown below:
  -------------------     ------------------
     Largest parts           Odd divisors
  -------------------     ------------------
   21   11    8    6       7    3   21    1
    |    |    |    |_______|    |    |    |
    |    |    |_________________|    |    |
    |    |___________________________|    |
    |_____________________________________|
.
So the 21st row of the triangle is [21, 11, 8, 6, 7, 3, 21, 1].
.
Illustration of initial terms in an isosceles triangle demonstrating the theorem described in A379630:
.                                          _ _
                                         _|1|1|_
                                       _|2 _|_ 1|_
                                     _|3  |2|3|  1|_
                                   _|4   _| | |_   1|_
                                 _|5    |3 _|_ 5|    1|_
                               _|6     _| |3|3| |_     1|_
                             _|7      |4  | | |  7|      1|_
                           _|8       _|  _| | |_  |_       1|_
                         _|9        |5  |4 _|_ 3|  9|        1|_
                       _|10        _|   | |4|5| |   |_         1|_
                     _|11         |6   _| | | | |_  11|          1|_
                   _|12          _|   |5  | | |  3|   |_           1|_
                 _|13           |7    |  _| | |_  |   13|            1|_
               _|14            _|    _| |5 _|_ 7| |_    |_             1|_
             _|15             |8    |6  | |5|5| |  3|   15|              1|_
           _|16              _|     |   | | | | |   |     |_               1|_
         _|17               |9     _|  _| | | | |_  |_    17|                1|_
       _|18                _|     |7  |6  | | |  9|  3|     |_                 1|_
     _|19                 |10     |   |  _| | |_  |   |     19|                  1|_
   _|20                  _|      _|   | |6 _|_ 5| |   |_      |_                   1|_
  |21                   |11     |8    | | |6|7| | |    3|     21|                    1|
.
The geometrical structure of the above isosceles triangle is defined in A237593. See also the triangles A286000 and A379633.
Note that the diagram also can be interpreted as a template which after folding gives a 90 degree pop-up card which has essentially the same structure as the stepped pyramid described in A245092.
.

Column 1 gives A000027.
Right border gives A000012.
The sum of row n equals A286015(n) + A000593(n).
The length of row n is A054844(n) = 2*A001227(n).
For another version with smallest parts see A379630.
The partitions of n into consecutive parts are in the n-th row of A299765. See also A286000.
The odd divisors of n are in the n-th row of A182469. See also A261697 and A261699.
Cf. A196020, A204217, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A262626, A379632, A379633, A379634.

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

A379288 Irregular triangle read by rows in which row n lists the odd divisors of n excluding odd divisors e for which there exists another divisor j with j < e < 2*j.

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A379631 Irregular triangle read by rows: T(n,m), n >= 1, m >= 1, in which row n lists the largest parts of the partitions of n into consecutive parts followed by the conjugate corresponding odd divisors of n in accordance with the theorem described in A379630.

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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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