A261697 -id:A261697 - 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.

A262626 Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593.

Original entry on oeis.org

1, 1, 1, 3, 2, 2, 2, 2, 2, 1, 1, 2, 7, 3, 1, 1, 3, 3, 3, 3, 2, 2, 3, 12, 4, 1, 1, 1, 1, 4, 4, 4, 4, 2, 1, 1, 2, 4, 15, 5, 2, 1, 1, 2, 5, 5, 3, 5, 5, 2, 2, 2, 2, 5, 9, 9, 6, 2, 1, 1, 1, 1, 2, 6, 6, 6, 6, 3, 1, 1, 1, 1, 3, 6, 28, 7, 2, 2, 1, 1, 2, 2, 7, 7, 7, 7, 3, 2, 1, 1, 2, 3, 7, 12, 12, 8, 3, 1, 2, 2, 1, 3, 8, 8, 8, 8, 8, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Sep 26 2015

Also the rows of both triangles A237270 and A237593 interleaved.
Also, irregular triangle read by rows in which T(n,k) is the area of the k-th region (from left to right in ascending diagonal) of the n-th symmetric set of regions (from the top to the bottom in descending diagonal) in the two-dimensional diagram of the perspective view of the infinite stepped pyramid described in A245092 (see the diagram in the Links section).
The diagram of the symmetric representation of sigma is also the top view of the pyramid, see Links section. For more information about the diagram see also A237593 and A237270.
The number of cubes at the n-th level is also A024916(n), the sum of all divisors of all positive integers <= n.
Note that this pyramid is also a quarter of the pyramid described in A244050. Both pyramids have infinitely many levels.
Odd-indexed rows are also the rows of the irregular triangle A237270.
Even-indexed rows are also the rows of the triangle A237593.
Lengths of the odd-indexed rows are in A237271.
Lengths of the even-indexed rows give 2*A003056.
Row sums of the odd-indexed rows gives A000203, the sum of divisors function.
Row sums of the even-indexed rows give the positive even numbers (see A005843).
Row sums give A245092.
From the front view of the stepped pyramid emerges a geometric pattern which is related to A001227, the number of odd divisors of the positive integers.
The connection with the odd divisors of the positive integers is as follows: A261697 --> A261699 --> A237048 --> A235791 --> A237591 --> A237593 --> A237270 --> this sequence.

			Irregular triangle begins:
  1;
  1, 1;
  3;
  2, 2;
  2, 2;
  2, 1, 1, 2;
  7;
  3, 1, 1, 3;
  3, 3;
  3, 2, 2, 3;
  12;
  4, 1, 1, 1, 1, 4;
  4, 4;
  4, 2, 1, 1, 2, 4;
  15;
  5, 2, 1, 1, 2, 5;
  5, 3, 5;
  5, 2, 2, 2, 2, 5;
  9, 9;
  6, 2, 1, 1, 1, 1, 2, 6;
  6, 6;
  6, 3, 1, 1, 1, 1, 3, 6;
  28;
  7, 2, 2, 1, 1, 2, 2, 7;
  7, 7;
  7, 3, 2, 1, 1, 2, 3, 7;
  12, 12;
  8, 3, 1, 2, 2, 1, 3, 8;
  8, 8, 8;
  8, 3, 2, 1, 1, 1, 1, 2, 3, 8;
  31;
  9, 3, 2, 1, 1, 1, 1, 2, 3, 9;
  ...
Illustration of the odd-indexed rows of triangle as the diagram of the symmetric representation of sigma which is also the top view of the stepped pyramid:
.
   n  A000203    A237270    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
   1     1   =      1      |_| | | | | | | | | | | | | | | |
   2     3   =      3      |_ _|_| | | | | | | | | | | | | |
   3     4   =    2 + 2    |_ _|  _|_| | | | | | | | | | | |
   4     7   =      7      |_ _ _|    _|_| | | | | | | | | |
   5     6   =    3 + 3    |_ _ _|  _|  _ _|_| | | | | | | |
   6    12   =     12      |_ _ _ _|  _| |  _ _|_| | | | | |
   7     8   =    4 + 4    |_ _ _ _| |_ _|_|    _ _|_| | | |
   8    15   =     15      |_ _ _ _ _|  _|     |  _ _ _|_| |
   9    13   =  5 + 3 + 5  |_ _ _ _ _| |      _|_| |  _ _ _|
  10    18   =    9 + 9    |_ _ _ _ _ _|  _ _|    _| |
  11    12   =    6 + 6    |_ _ _ _ _ _| |  _|  _|  _|
  12    28   =     28      |_ _ _ _ _ _ _| |_ _|  _|
  13    14   =    7 + 7    |_ _ _ _ _ _ _| |  _ _|
  14    24   =   12 + 12   |_ _ _ _ _ _ _ _| |
  15    24   =  8 + 8 + 8  |_ _ _ _ _ _ _ _| |
  16    31   =     31      |_ _ _ _ _ _ _ _ _|
  ...
The above diagram arises from a simpler diagram as shown below.
Illustration of the even-indexed rows of triangle as the diagram of the deployed front view of the corner of the stepped pyramid:
.
.                                 A237593
Level                               _ _
1                                 _|1|1|_
2                               _|2 _|_ 2|_
3                             _|2  |1|1|  2|_
4                           _|3   _|1|1|_   3|_
5                         _|3    |2 _|_ 2|    3|_
6                       _|4     _|1|1|1|1|_     4|_
7                     _|4      |2  |1|1|  2|      4|_
8                   _|5       _|2 _|1|1|_ 2|_       5|_
9                 _|5        |2  |2 _|_ 2|  2|        5|_
10              _|6         _|2  |1|1|1|1|  2|_         6|_
11            _|6          |3   _|1|1|1|1|_   3|          6|_
12          _|7           _|2  |2  |1|1|  2|  2|_           7|_
13        _|7            |3    |2 _|1|1|_ 2|    3|            7|_
14      _|8             _|3   _|1|2 _|_ 2|1|_   3|_             8|_
15    _|8              |3    |2  |1|1|1|1|  2|    3|              8|_
16   |9                |3    |2  |1|1|1|1|  2|    3|                9|
...
The number of horizontal line segments in the n-th level in each side of the diagram equals A001227(n), the number of odd divisors of n.
The number of horizontal line segments in the left side of the diagram plus the number of the horizontal line segment in the right side equals A054844(n).
The total number of vertical line segments in the n-th level of the diagram equals A131507(n).
The diagram represents the first 16 levels of the pyramid.
The diagram of the isosceles triangle and the diagram of the top view of the pyramid shows the connection between the partitions into consecutive parts and the sum of divisors function (see also A286000 and A286001). - _Omar E. Pol_, Aug 28 2018
The connection between the isosceles triangle and the stepped pyramid is due to the fact that this object can also be interpreted as a pop-up card. - _Omar E. Pol_, Nov 09 2022

Robert Price, Table of n, a(n) for n = 1..16048 (n=1..412 rows)
Omar E. Pol, An infinite stepped pyramid (A237593, A237270, A262626)
Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Perspective view of the stepped pyramid (levels: 1..16)
Index entries for sequences related to sigma(n)

Famous sequences that are visible in the stepped pyramid:
Cf. A000040 (prime numbers)......., for the characteristic shape see A346871.
Cf. A000079 (powers of 2)........., for the characteristic shape see A346872.
Cf. A000203 (sum of divisors)....., total area of the terraces in the n-th level.
Cf. A000217 (triangular numbers).., for the characteristic shape see A346873.
Cf. A000225 (Mersenne numbers)...., for a visualization see A346874.
Cf. A000384 (hexagonal numbers)..., for the characteristic shape see A346875.
Cf. A000396 (perfect numbers)....., for the characteristic shape see A346876.
Cf. A000668 (Mersenne primes)....., for a visualization see A346876.
Cf. A001097 (twin primes)........., for a visualization see A346871.
Cf. A001227 (# of odd divisors)..., number of subparts in the n-th level.
Cf. A002378 (oblong numbers)......, for a visualization see A346873.
Cf. A008586 (multiples of 4)......, perimeters of the successive levels.
Cf. A008588 (multiples of 6)......, for the characteristic shape see A224613.
Cf. A013661 (zeta(2))............., (area of the horizontal faces)/(n^2), n -> oo.
Cf. A014105 (second hexagonals)..., for the characteristic shape see A346864.
Cf. A067742 (# of middle divisors), # cells in the main diagonal in n-th level.
Other sequences that are visible in the stepped pyramid: A000096, A001065, A001359, A001747, A002939, A002943, A003056, A004125, A004277, A004526, A005279, A006512, A007606, A007607, A082647, A008438, A008578, A008864, A010814, A014106, A014107, A014132, A014574, A016945, A019434, A024206, A024916, A028552, A028982, A028983, A034856, A038550, A047836, A048050, A052928, A054735, A054844, A062731, A065091, A065475, A071561, A071562, A071904, A092506, A100484, A108605, A139256, A139257, A144396, A152677, A152678, A153485, A155085, A161680, A161983, A162917, A174905, A174973, A175254, A176810, A224880, A235791, A237270, A237271, A237591, A237593, A238005, A238524, A244049, A245092, A259176, A259177, A261348, A278972, A317302, A317303, A317304, A317305, A317307, A319529, A319796, A319801, A319802, A327329, A336305, (and several others).
Apart from zeta(2) other constants that are related to the stepped pyramid are A072691, A353908, A354238.
Cf. A054844, A131507, A196020, A236104, A237048, A239660, A244050, A259179, A261350, A261697, A261699, A262612, A280850, A286000, A286001, A296508.

A261699 Triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists positive terms interleaved with k-1 zeros, starting in row k(k+1)/2. If k is odd the positive terms of column k are k's, otherwise if k is even the positive terms of column k are the odd numbers greater than k in increasing order.

Original entry on oeis.org

1, 1, 1, 3, 1, 0, 1, 5, 1, 0, 3, 1, 7, 0, 1, 0, 0, 1, 9, 3, 1, 0, 0, 5, 1, 11, 0, 0, 1, 0, 3, 0, 1, 13, 0, 0, 1, 0, 0, 7, 1, 15, 3, 0, 5, 1, 0, 0, 0, 0, 1, 17, 0, 0, 0, 1, 0, 3, 9, 0, 1, 19, 0, 0, 0, 1, 0, 0, 0, 5, 1, 21, 3, 0, 0, 7, 1, 0, 0, 11, 0, 0, 1, 23, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 1, 25, 0, 0, 5, 0, 1, 0, 0, 13, 0, 0

Offset: 1

Omar E. Pol, Sep 20 2015

Conjecture: the positive terms in row n are the odd divisors of n.
Note that the elements appear with an unusual ordering, for example; row 45 is 1, 45, 3, 0, 5, 15, 0, 0, 9.
The positive terms give A261697.
Row n has length A003056(n) hence column k starts in row A000217(k).
The number of positive terms in row n is A001227(n).
The sum of row n is A000593(n).
The connection with the symmetric representation of sigma is as follows: A237048 --> A235791 --> A237591 --> A237593.
Proof of the conjecture: let n = 2^m*s*t with s and t odd. The property stated in A237048 verifies the conjecture with odd divisor k <= A003056(n) of n in position k and odd divisor t > A003056(n) in position 2^(m+1)*s. Therefore reading in row n the nonzero odd positions from left to right and then the nonzero even positions from right to left gives all odd divisors of n in increasing order. - Hartmut F. W. Hoft, Oct 25 2015
A237048 gives the signum function (A057427) of this sequence. - Omar E. Pol, Nov 14 2016
From Peter Munn, Jul 30 2017: (Start)
Each odd divisor d of n corresponds to n written as a sum of consecutive integers (n/d - (d-1)/2) .. (n/d + (d-1)/2). After canceling any corresponding negative and positive terms and deleting any zero term, the lower bound becomes abs(n/d - d/2) + 1/2, leaving k terms where k = n/d + d/2 - abs(n/d - d/2). It can be shown T(n,k) = d.
This sequence thereby defines a one to one relationship between odd divisors of n and partitions of n into k consecutive parts.
The relationship is expressed below using 4 sequences (with matching row lengths), starting with this one:
A261699(n,k) = d, the odd divisor.
A211343(n,k) = abs(n/d - d/2) + 1/2, smallest part.
A285914(n,k) = k, number of parts.
A286013(n,k) = n/d + (d-1)/2, largest part.
If no partition of n into k consecutive parts exists, the corresponding sequence terms are 0.
(End)

			Triangle begins:
1;
1;
1,  3;
1,  0;
1,  5;
1,  0,  3;
1,  7,  0;
1,  0,  0;
1,  9,  3;
1,  0,  0,  5;
1, 11,  0,  0;
1,  0,  3,  0;
1, 13,  0,  0;
1,  0,  0,  7;
1, 15,  3,  0,  5;
1,  0,  0,  0,  0;
1, 17,  0,  0,  0;
1,  0,  3,  9,  0;
1, 19,  0,  0,  0;
1,  0,  0,  0,  5;
1, 21,  3,  0,  0,  7;
1,  0,  0, 11,  0,  0;
1, 23,  0,  0,  0,  0;
1,  0,  3,  0,  0,  0;
1, 25,  0,  0,  5,  0;
1,  0,  0, 13,  0,  0;
1, 27,  3,  0,  0,  9;
1,  0,  0,  0,  0,  0,  7;
...
From _Omar E. Pol_, Dec 19 2016: (Start)
Illustration of initial terms in a right triangle whose structure is the same as the structure of A237591:
Row                                                         _
1                                                         _|1|
2                                                       _|1 _|
3                                                     _|1  |3|
4                                                   _|1   _|0|
5                                                 _|1    |5 _|
6                                               _|1     _|0|3|
7                                             _|1      |7  |0|
8                                           _|1       _|0 _|0|
9                                         _|1        |9  |3 _|
10                                      _|1         _|0  |0|5|
11                                    _|1          |11  _|0|0|
12                                  _|1           _|0  |3  |0|
13                                _|1            |13   |0 _|0|
14                              _|1             _|0   _|0|7 _|
15                            _|1              |15   |3  |0|5|
16                          _|1               _|0    |0  |0|0|
17                        _|1                |17    _|0 _|0|0|
18                      _|1                 _|0    |3  |9  |0|
19                    _|1                  |19     |0  |0 _|0|
20                  _|1                   _|0     _|0  |0|5 _|
21                _|1                    |21     |3   _|0|0|7|
22              _|1                     _|0      |0  |11 |0|0|
23            _|1                      |23      _|0  |0  |0|0|
24          _|1                       _|0      |3    |0 _|0|0|
25        _|1                        |25       |0   _|0|5  |0|
26      _|1                         _|0       _|0  |13 |0 _|0|
27    _|1                          |27       |3    |0  |0|9 _|
28   |1                            |0        |0    |0  |0|0|7|
... (End)

Cf. A000217, A000593, A001227, A003056, A005408, A027750, A057427, A182469, A196020, A211343, A236104, A235791, A236112, A237048, A237591, A237593, A261350, A261697, A261698, A285914, A286013.

T[n_, k_?OddQ] /; n == k (k + 1)/2 := k; T[n_, k_?OddQ] /; Mod[n - k (k + 1)/2, k] == 0 := k; T[n_, k_?EvenQ] /; n == k (k + 1)/2 := k + 1; T[n_, k_?EvenQ] /; Mod[n - k (k + 1)/2, k] == 0 := T[n - k, k] + 2; T[, ] = 0; Table[T[n, k], {n, 1, 26}, {k, 1, Floor[(Sqrt[1 + 8 n] - 1)/2]}] // Flatten (* Jean-François Alcover, Sep 21 2015 *)
(* alternate definition using function a237048 *)
T[n_, k_] := If[a237048[n, k] == 1, If[OddQ[k], k, 2n/k], 0] (* Hartmut F. W. Hoft, Oct 25 2015 *)

From Hartmut F. W. Hoft, Oct 25 2015: (Start)
T(n, k) = 2n/k, if A237048(n, k) = 1 and k even,
and in accordance with the definition:
T(n, k) = k, if A237048(n, k) = 1 and k odd,
T(n, k) = 0 otherwise; for k <= A003056(n).
(End)
For m >= 1, d >= 1 and odd, T(m*d, m + d/2 - abs(m - d/2)) = d. - Peter Munn, Jul 24 2017

A266531 Square array read by antidiagonals upwards: T(n,k) = n-th number with k odd divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 8, 6, 18, 15, 16, 7, 25, 21, 81, 32, 10, 36, 27, 162, 45, 64, 11, 49, 30, 324, 63, 729, 128, 12, 50, 33, 625, 75, 1458, 105, 256, 13, 72, 35, 648, 90, 2916, 135, 225, 512, 14, 98, 39, 1250, 99, 5832, 165, 441, 405, 1024, 17, 100, 42, 1296, 117, 11664, 189, 450, 567, 59049, 2048, 19, 121, 51, 2401, 126, 15625

Offset: 1

Omar E. Pol, Apr 02 2016

T(n,k) is the n-th positive integer with exactly k odd divisors.
This is a permutation of the natural numbers.
T(n,k) is also the n-th number j with the property that the symmetric representation of sigma(j) has k subparts (cf. A279387). - Omar E. Pol, Dec 27 2016
T(n,k) is also the n-th positive integer with exactly k partitions into consecutive parts. - Omar E. Pol, Aug 16 2018

			The corner of the square array begins:
    1,  3,  9, 15,   81,  45,   729, 105,  225,  405, ...
    2,  5, 18, 21,  162,  63,  1458, 135,  441,  567, ...
    4,  6, 25, 27,  324,  75,  2916, 165,  450,  810, ...
    8,  7, 36, 30,  625,  90,  5832, 189,  882,  891, ...
   16, 10, 49, 33,  648,  99, 11664, 195,  900, 1053, ...
   32, 11, 50, 35, 1250, 117, 15625, 210, 1089, 1134, ...
   64, 12, 72, 39, 1296, 126, 23328, 231, 1225, 1377, ...
  128, 13, 98, 42, 2401, 147, 31250, 255, 1521, 1539, ...
  ...

Index entries for sequences that are permutations of the natural numbers

Row 1 is A038547.
Column 1-10: A000079, A038550, A072502, apparently A131651, A267696, A230577, A267697, A267891, A267892, A267893.
Cf. A001227, A182469, A236104, A237591, A237593, A240062, A261697, A261698, A261699, A279387, A286000, A286001, A296508.

A379630 Irregular triangle read by rows in which row n lists the smallest parts of the partitions of n into consecutive parts followed by the conjugate corresponding odd divisors of n in accordance with the theorem of correspondence described in the Comments lines.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 28 2024

Theorem of correspondence between the partitions of n into k consecutive parts and the odd divisors of n: given a partition of n into k consecutive parts if k is odd then the corresponding odd divisor of n is k, otherwise if k is even then the corresponding odd divisor of n is the sum of any pair of conjugate parts of the partition (for example the sum of the largest part and the smallest part).
Conjecture: the first A001227(n) terms in the n-th row are also the absolute values of the n-th row of A341971.
The last A001227(n) terms in the n-th row are also the mirror of the n-th row of A261697.

			Triangle begins:
   1,  1;
   2,  1;
   3,  1,  3,  1;
   4,  1;
   5,  2,  5,  1;
   6,  1,  3,  1;
   7,  3,  7,  1;
   8,  1;
   9,  4,  2,  3,  9,  1;
  10,  1,  5,  1;
  11,  5, 11,  1;
  12,  3,  3,  1;
  13,  6, 13,  1;
  14,  2,  7,  1;
  15,  7,  4,  1,  5,  3, 15,  1;
  16,  1;
  17,  8, 17,  1;
  18,  5,  3,  9,  3,  1;
  19,  9, 19,  1;
  20,  2,  5,  1;
  21, 10,  6,  1,  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 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 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 smallest parts at the top as shown below:
  ------------------------------------------
           Conjugate correspondence
  ------------------------------------------
    Partitions of 21              Odd
    into consecutive           divisors
    parts as columns             of 21
  -------------------     ------------------
   21   10    6    1       7    3   21    1
    |   11    7    2       |    |    |    |
    |    |    8    3       |    |    |    |
    |    |    |    4       |    |    |    |
    |    |    |    5       |    |    |    |
    |    |    |    6       |    |    |    |
    |    |    |    |_______|    |    |    |
    |    |    |_________________|    |    |
    |    |___________________________|    |
    |_____________________________________|
.
Then removing all rows except the first row we have a table of conjugate correspondence for smallest parts and odd divisors as shown below:
  -------------------     ------------------
    Smallest parts           Odd divisors
  -------------------     ------------------
   21   10    6    1       7    3   21    1
    |    |    |    |_______|    |    |    |
    |    |    |_________________|    |    |
    |    |___________________________|    |
    |_____________________________________|
.
So the 21st row of the triangle is [21, 10, 6, 1, 7, 3, 21, 1].
.
Illustration of initial terms in an isosceles triangle demonstrating the theorem:
.                                          _ _
                                         _|1|1|_
                                       _|2 _|_ 1|_
                                     _|3  |1|3|  1|_
                                   _|4   _| | |_   1|_
                                 _|5    |2 _|_ 5|    1|_
                               _|6     _| |1|3| |_     1|_
                             _|7      |3  | | |  7|      1|_
                           _|8       _|  _| | |_  |_       1|_
                         _|9        |4  |2 _|_ 3|  9|        1|_
                       _|10        _|   | |1|5| |   |_         1|_
                     _|11         |5   _| | | | |_  11|          1|_
                   _|12          _|   |3  | | |  3|   |_           1|_
                 _|13           |6    |  _| | |_  |   13|            1|_
               _|14            _|    _| |2 _|_ 7| |_    |_             1|_
             _|15             |7    |4  | |1|5| |  3|   15|              1|_
           _|16              _|     |   | | | | |   |     |_               1|_
         _|17               |8     _|  _| | | | |_  |_    17|                1|_
       _|18                _|     |5  |3  | | |  9|  3|     |_                 1|_
     _|19                 |9      |   |  _| | |_  |   |     19|                  1|_
   _|20                  _|      _|   | |2 _|_ 5| |   |_      |_                   1|_
  |21                   |10     |6    | | |1|7| | |    3|     21|                    1|
.
The geometrical structure of the above isosceles triangle is defined in A237593. See also the triangle A286001.
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 A286014(n) + A000593(n).
The length of row n is A054844(n) = 2*A001227(n).
The partitions of n into consecutive parts are in the n-th row of A299765. See also A286000 and A286001.
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, A341971.

A261698 Irregular triangle read by rows in which row n lists the odd divisors of n in the order as follows: the smallest, the largest, the second smallest, the second largest, the third smallest, the third largest, and so on.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 5, 1, 3, 1, 7, 1, 1, 9, 3, 1, 5, 1, 11, 1, 3, 1, 13, 1, 7, 1, 15, 3, 5, 1, 1, 17, 1, 9, 3, 1, 19, 1, 5, 1, 21, 3, 7, 1, 11, 1, 23, 1, 3, 1, 25, 5, 1, 13, 1, 27, 3, 9, 1, 7, 1, 29, 1, 15, 3, 5, 1, 31, 1, 1, 33, 3, 11, 1, 17, 1, 35, 5, 7, 1, 9, 3, 1, 37, 1, 19, 1, 39, 3, 13, 1, 5, 1, 41, 1, 21, 3, 7, 1, 43, 1, 11, 1, 45, 3, 15, 5, 9

Offset: 1

Omar E. Pol, Sep 21 2015

Also the odd numbers of A210959.
Row lengths give A001227.
Row sums give A000593.
Another version of A182469 from which differs at a(14), or T(9,2).
For a similar version see A261697 from which differs at a(34), or T(18,2).

			List of divisors of 45 from distinct sequences:
45th row of triangle A182469: 1, 3, 5, 9, 15, 45.
45th row of triangle A261697: 1, 45, 3, 5, 15, 9.
45th row of this triangle...: 1, 45, 3, 15, 5, 9.
Triangle begins:
  1;
  1;
  1,  3;
  1;
  1,  5;
  1,  3;
  1,  7;
  1;
  1,  9,  3;
  1,  5;
  1, 11;
  1,  3;
  1, 13;
  1,  7;
  1, 15,  3,  5;
  1;
  1, 17;
  1,  9,  3;
  1, 19;
  1,  5;
  1, 21,  3,  7;
  ...

Cf. A000593, A001227, A027750, A182469, A210959, A261697.

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.

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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A262626 Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593.

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A266531 Square array read by antidiagonals upwards: T(n,k) = n-th number with k odd divisors.

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A379630 Irregular triangle read by rows in which row n lists the smallest parts of the partitions of n into consecutive parts followed by the conjugate corresponding odd divisors of n in accordance with the theorem of correspondence described in the Comments lines.

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A261698 Irregular triangle read by rows in which row n lists the odd divisors of n in the order as follows: the smallest, the largest, the second smallest, the second largest, the third smallest, the third largest, and so on.

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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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