A182469 -id:A182469 - cp's OEIS frontend

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.

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.

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

Original entry on oeis.org

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.

A383402 Smallest number whose largest odd divisor is its n-th divisor.

Original entry on oeis.org

1, 3, 6, 15, 18, 36, 30, 105, 60, 120, 90, 315, 816, 1360, 180, 700, 450, 360, 720, 1008, 420, 1540, 630, 900, 840, 1080, 1620, 1680, 2160, 1800, 1890, 5280, 1260, 3240, 3150, 17325, 7200, 29120, 5670, 9072, 2520, 3960, 10296, 18144, 3780, 20020, 5040, 7920, 10800

Offset: 1

Omar E. Pol, May 14 2025

nonn

From Peter Munn, May 15 2025 and May 20 2025: (Start)
A038547 is easily seen to be an upper bound for the sequence and a term equals this upper bound if and only if it is odd. Moreover, if a(n) = 2m with m odd, then the largest odd divisor of 2m is m, its second largest divisor, and a(n) = 2 * A038547((n+1)/2). It follows that 1 is the only term not divisible by 4 or by a nonunit term of A038547.
a(8) = 105 is the last squarefree term. (This is a corollary to lemma: prime p > 9 cannot be a divisor of a squarefree term. Proof of lemma: Let p divide squarefree k. If 3p is also divisor, set m = 9k/p, otherwise set m = 3k/p. Then k is not a term as m is a smaller number whose largest odd divisor is in the same position in the divisor list.)
(End)
If a(n) = m then m has at least n divisors. - David A. Corneth, May 16 2025
Every term a(n) = t > 1 is divisible by 2 or 3. Proof: Suppose it is not. Then it is odd and n is the number of divisors of t (cf. A000005). But t is not the smallest number that has n odd divisors that is odd. Setting every prime factor p to the largest prime < p and then multiplying gives a smaller odd number that has n divisors (cf. A064989). - David A. Corneth, May 17 2025

			The divisors of 18 are [1, 2, 3, 6, 9, 18] and the largest odd divisor is 9 and 9 is its 5th divisor, so a(5) = 18 because 18 the smallest number having that property.

David A. Corneth, Table of n, a(n) for n = 1..872
David A. Corneth, PARI program
David A. Corneth, Upper bounds on a(n) for n = 1..10000
Michael De Vlieger, Prime Power Decomposition of A383402(n), n = 1..261.

Row 1 of A383961.
The range of terms is a subset of {1} U A355200.
Cf. A000005, A000265, A001227, A005117, A027750, A038547, A064989, A182469, A383401.
See A221647 for other sequences giving the smallest number whose n-th divisor satisfies some condition.

With[{t = Table[If[OddQ[n], DivisorSigma[0, n], FirstPosition[Divisors[n], n/2^IntegerExponent[n, 2]][[1]]], {n, 1, 30000}]}, TakeWhile[FirstPosition[t, #] & /@ Range[Max[t]] // Flatten, ! MissingQ[#] &]] (* Amiram Eldar, May 14 2025 *)

a(n) = my(k=1); while (select(x->(x==k/2^valuation(k,2)), divisors(k), 1)[1] != n, k++); k; \\ Michel Marcus, May 14 2025

PARI
```
\\ See Corneth link
```

a(n) = min({k : A000005(k) >= n & A027750(k,n) = A000265(k)}). - Peter Munn, May 14 2025

More terms from Amiram Eldar, May 14 2025

A285574 Irregular triangle read by rows which arises from a diagram which is similar to the diagram of A261699, but here the even-indexed zig-zag lines are located on the right-hand side of the vertical axis of the diagram.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Apr 21 2017

In the diagram we have that:
The number of horizontal line segments in the n-th row of the structure equals A001227(n), the number of partitions of n into consecutive parts.
The number of horizontal line segments in the left-hand part of the n-th row equals A082647, the number of partitions of n into an odd number of consecutive parts.
The number of horizontal line segments in the right-hand part of the n-th row equals A131576, the number of partitions of n into an even number of consecutive parts.
The diagram explains the unusual ordering of the terms in the triangle A261699, in which the even-indexed zig-zag lines are located on the left-hand side of the vertical axis of the diagram.

			Triangle begins:
1;
1;
1, 3;
1, 0,
1, 5;
1, 3, 0;
1, 0, 7;
1, 0, 0;
1, 3, 9;
1, 0, 5,  0;
1, 0, 0, 11;
1, 3, 0,  0;
1, 0, 0, 13;
1, 0, 7,  0;
1, 3, 5,  0, 15;
1, 0, 0,  0,  0;
1, 0, 0,  0, 17;
1, 3, 0,  9,  0;
1, 0, 0,  0, 19;
1, 0, 5,  0,  0;
1, 3, 0,  7,  0, 21;
...
Illustration of initial terms of the diagram:
Row                                           _
1                                           _|1|
2                                         _|1  |_
3                                       _|1    |3|
4                                     _|1      |0|_
5                                   _|1       _|  5|
6                                 _|1        |3|  0|_
7                               _|1          |0|    7|
8                             _|1           _|0|    0|_
9                           _|1            |3  |_     9|
10                        _|1              |0  |5|    0|_
11                      _|1               _|0  |0|     11|
12                    _|1                |3    |0|      0|_
13                  _|1                  |0    |0|_      13|
14                _|1                   _|0   _|  7|      0|_
15              _|1                    |3    |5|  0|       15|
16            _|1                      |0    |0|  0|        0|_
17          _|1                       _|0    |0|  0|_        17|
18        _|1                        |3      |0|    9|        0|_
19      _|1                          |0     _|0|    0|         19|
20    _|1                           _|0    |5  |_   0|          0|_
21   |1                            |3      |0  |7|  0|           21|
...
(Compare with the diagram of A261699.)

Positive terms give A182469.
Row n has length A003056(n).
The sum of row n is A000593(n).
Column k starts in row A000217(k).
The number of positive terms in row n is A001227(n).
Cf. A082647, A131576, A196020, A235791, A236104, A237048, A237591, A237593, A245092, A259176, A259177, A261699, A279820.

A379461 Irregular triangle read by rows in which row n lists the divisors m of n such that there is a divisor d of n with d < m < 2*d, or 0 if such divisors do not exist.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 4, 6, 0, 0, 5, 0, 0, 3, 9, 0, 5, 0, 0, 0, 3, 4, 6, 8, 12, 0, 0, 0, 7, 0, 3, 5, 6, 10, 15, 0, 0, 0, 0, 7, 3, 4, 6, 9, 12, 18, 0, 0, 0, 5, 8, 10, 0, 3, 7, 21, 0, 0, 5, 9, 15, 0, 0, 3, 4, 6, 8, 12, 16, 24, 0, 0, 0, 0, 0, 3, 9, 27, 0

Offset: 1

Omar E. Pol, Dec 23 2024

The number of positive terms in row n is A174903(n).
The indices of the rows that contain a zero give A174905.
The indices of the rows that contain positive integers give A005279.
The positive integers in the n-th row are the missing divisors of n in the n-th row of A379374.
The odd integers in the n-th row are the missing odd divisors of n in the n-th row of A379288.

			Triangle begins:
  0;
  0;
  0;
  0;
  0;
  3;
  0;
  0;
  0;
  0;
  0;
  3, 4, 6;
  0;
  0;
  5;
  0;
  0;
  3, 9;
  0;
  5;
  ...
From _Omar E. Pol_, Apr 19 2025: (Start)
For n = 12 there are three divisors m of 12 such that there is a divisor d of 12 with d < m < 2*d. Those divisors are 3, 4 and 6 as shown below:
   d  <  m  <  2*d
--------------------
   1            2
   2     3      4
   3     4      6
   4     6      8
   6           12
  12           24
.
So the 12th row of the triangle is [3, 4, 6]. (End)

Cf. A005279, A027750, A174903, A174905, A182469, A237271, A379288, A379374, A379379, A379384, A383209.

row[n_] := Module[{d = Partition[Divisors[n], 2, 1], e}, e = Select[d, #[[2]] < 2*#[[1]] &][[;; , 2]]; If[e == {}, {0}, e]]; Table[row[n], {n, 1, 55}] // Flatten (* Amiram Eldar, Dec 23 2024 *)

More terms from Amiram Eldar, Dec 23 2024

Name changed by Omar E. Pol, Feb 05 2025

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.

A383961 Square array read by upward antidiagonals: T(n,k) is the n-th number whose largest odd divisor is its k-th divisor, n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 9, 15, 16, 11, 10, 20, 18, 32, 13, 12, 21, 50, 36, 64, 17, 14, 27, 81, 45, 30, 128, 19, 22, 28, 88, 63, 42, 105, 256, 23, 24, 33, 98, 75, 54, 135, 60, 512, 29, 25, 35, 104, 99, 66, 165, 84, 120, 1024, 31, 26, 39, 136, 117, 70, 189, 108, 140, 90

Offset: 1

Omar E. Pol, May 16 2025

This is a permutation of the positive integers.
From Peter Munn, May 18 2025: (Start)
Numbers with the same factorization pattern of their sequence of divisors (see A290110) and the same parity appear here in the same column.
For example, each column k > 2 includes the subsequence 2^(k-2) * p for all prime p > 2^(k-2).
(End)

			The corner 15 X 15 of the square array is as follows:
      1,  3,  6,  15,  18,  36,  30, 105,  60, 120,  90, 315,  816, 1360, 180, ...
      2,  5,  9,  20,  50,  45,  42, 135,  84, 140, 126, 324,  880, 1520, 210, ...
      4,  7, 10,  21,  81,  63,  54, 165, 108, 168, 150, 432,  912, 1632, 252, ...
      8, 11, 12,  27,  88,  75,  66, 189, 132, 220, 198, 440, 1040, 1760, 270, ...
     16, 13, 14,  28,  98,  99,  70, 195, 156, 240, 216, 495, 1056, 1824, 300, ...
     32, 17, 22,  33, 104, 117,  72, 200, 162, 260, 234, 520, 1104, 1840, 330, ...
     64, 19, 24,  35, 136, 147,  78, 231, 204, 308, 264, 525, 1120, 1904, 378, ...
    128, 23, 25,  39, 152, 153, 100, 255, 225, 340, 280, 528, 1144, 2000, 390, ...
    256, 29, 26,  40, 176, 171, 102, 273, 228, 364, 294, 560, 1232, 2080, 396, ...
    512, 31, 34,  44, 184, 175, 110, 285, 276, 380, 306, 585, 1248, 2128, 462, ...
   1024, 37, 38,  51, 208, 207, 114, 297, 348, 405, 312, 616, 1392, 2208, 468, ...
   2048, 41, 46,  52, 232, 243, 130, 345, 372, 460, 336, 624, 1456, 2288, 510, ...
   4096, 43, 48,  55, 242, 245, 138, 351, 400, 476, 342, 675, 1458, 2320, 546, ...
   8192, 47, 49,  56, 248, 261, 144, 357, 441, 480, 350, 680, 1488, 2464, 570, ...
  16384, 53, 58,  57, 296, 272, 154, 375, 444, 500, 408, 693, 1496, 2480, 588, ...
  ...

Column 1 gives A000079.
Column 2 gives A065091.
Column 3 consists of (A001248 U A091629 U A100484)\{4}.
Column 4 consists of numbers >= 15 in (A001749 U A030078 U A046388 U A070875).
Row 1 gives A383402.
Cf. A000005, A000265, A001227, A027750, A038547, A174090, A182469, A290110, A383401.

f[n_] := If[OddQ[n], DivisorSigma[0, n], FirstPosition[Divisors[n], n/2^IntegerExponent[n, 2]][[1]]]; seq[m_] := Module[{t = Table[0, {m}, {m}], v = Table[0, {m}], c = 0, k = 1, i, j}, While[c < m*(m + 1)/2, i = f[k]; If[i <= m, j = v[[i]] + 1; If[j <= m - i + 1, t[[i]][[j]] = k; v[[i]]++; c++]]; k++]; Table[t[[j]][[i - j + 1]], {i, 1, m}, {j, 1, i}] // Flatten]; seq[11] (* Amiram Eldar, May 16 2025 *)

A383962 Irregular triangle read by rows: T(n,k) is the index of the k-th odd divisor in the list of divisors of n, with n >= 1, k >= 1.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, May 26 2025

Row n lists the indices of the odd divisors in the list of divisors of n.
If n is odd then row n lists the first A000005(n) positive integers (A000027).
Row n is [1] if and only if n is a power of 2 (A000079).
Row n is [1, 2] if and only if n is an odd prime (A065091).

			Triangle begins (n = 1..21):
  1;
  1;
  1, 2;
  1;
  1, 2;
  1, 3;
  1, 2;
  1;
  1, 2, 3;
  1, 3;
  1, 2;
  1, 3;
  1, 2;
  1, 3;
  1, 2, 3, 4;
  1;
  1, 2;
  1, 3, 5;
  1, 2;
  1, 4;
  1, 2, 3, 4;
  ...
For n = 20 the divisors of 20 are [1, 2, 4, 5, 10, 20]. The odd divisors are [1, 5] and their indices in the list of divisors are [1, 4] respectively, so the 20th row of the triangle is [1, 4].

Column 1 gives A000012.
Row lengths gives A001227.
Right border gives A383401.
Cf. A000005, A000027, A000079, A027750, A065091, A174090, A182469.

row[n_] := Position[Divisors[n], ?OddQ] // Flatten; Array[row, 45] // Flatten (* _Amiram Eldar, May 26 2025 *)

A290514 Numbers n such that product of odd divisors of n > product of odd divisors of m for all m < n.

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 21, 27, 33, 35, 39, 45, 63, 75, 99, 105, 135, 165, 189, 195, 225, 315, 495, 525, 585, 675, 693, 735, 765, 819, 825, 855, 945, 1155, 1365, 1485, 1575, 2205, 2475, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 7425, 7875, 8085, 8415, 8505, 8775, 8925, 9009, 9135, 9405

Offset: 1

Ilya Gutkovskiy, Aug 04 2017

nonn

Numbers n such that A136655(n) > A136655(m) for all m < n.

Cf. A034287, A034288, A136655, A174572, A182469.

mx = 0; t = {}; Do[u = Product[d, {d, Select[Divisors[n], OddQ[#] &]}]; If[u > mx, mx = u; AppendTo[t, n]], {n, 9500}]; t

cp's OEIS Frontend

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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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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A383402 Smallest number whose largest odd divisor is its n-th divisor.

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PARI

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A285574 Irregular triangle read by rows which arises from a diagram which is similar to the diagram of A261699, but here the even-indexed zig-zag lines are located on the right-hand side of the vertical axis of the diagram.

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A379461 Irregular triangle read by rows in which row n lists the divisors m of n such that there is a divisor d of n with d < m < 2*d, or 0 if such divisors do not exist.

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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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A383961 Square array read by upward antidiagonals: T(n,k) is the n-th number whose largest odd divisor is its k-th divisor, n >= 1, k >= 1.

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A383962 Irregular triangle read by rows: T(n,k) is the index of the k-th odd divisor in the list of divisors of n, with n >= 1, k >= 1.

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A290514 Numbers n such that product of odd divisors of n > product of odd divisors of m for all m < n.

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