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
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.
.
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.
Cf.
A196020,
A204217,
A235791,
A236104,
A237048,
A237270,
A237271,
A237591,
A237593,
A245092,
A262626,
A379632,
A379633,
A379634.
A384930
Irregular triangle read by rows: T(n,k) is the sum of the terms of the (n-k+1)-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.
Original entry on oeis.org
1, 3, 3, 1, 7, 5, 1, 12, 7, 1, 15, 9, 3, 1, 15, 3, 11, 1, 28, 13, 1, 21, 3, 15, 8, 1, 31, 17, 1, 39, 19, 1, 42, 21, 7, 3, 1, 33, 3, 23, 1, 60, 25, 5, 1, 39, 3, 27, 9, 3, 1, 56, 29, 1, 72, 31, 1, 63, 33, 11, 3, 1, 51, 3, 35, 12, 1, 91, 37, 1, 57, 3, 39, 13, 3, 1, 90, 41, 1, 96, 43, 1, 77, 7, 45, 32, 1
Offset: 1
---------------------------------------------------------------------
| n | Row n of | List of divisors of n | Number of |
| | the triangle | [with sublists in brackets] | sublists |
---------------------------------------------------------------------
| 1 | 1; | [1]; | 1 |
| 2 | 3; | [1, 2]; | 1 |
| 3 | 3, 1; | [1], [3]; | 2 |
| 4 | 7; | [1, 2, 4]; | 1 |
| 5 | 5, 1; | [1], [5]; | 2 |
| 6 | 12; | [1, 2, 3, 6]; | 1 |
| 7 | 7, 1; | [1], [7]; | 2 |
| 8 | 15; | [1, 2, 4, 8]; | 1 |
| 9 | 9, 3, 1; | [1], [3], [9]; | 3 |
| 10 | 15 3; | [1, 2], [5, 10]; | 2 |
| 11 | 11, 1; | [1], [11]; | 2 |
| 12 | 28; | [1, 2, 3, 4, 6, 12]; | 1 |
| 13 | 13, 1; | [1], [13]; | 2 |
| 14 | 21, 3; | [1, 2], [7, 14]; | 2 |
| 15 | 15, 8, 1; | [1], [3, 5], [15]; | 3 |
| 16 | 31; | [1, 2, 4, 8, 16]; | 1 |
| 17 | 17, 1; | [1], [17]; | 2 |
| 18 | 39; | [1, 2, 3, 6, 9, 18]; | 1 |
| 19 | 19, 1; | [1], [19]; | 2 |
| 20 | 42; | [1, 2, 4, 5, 10, 20]; | 1 |
| 21 | 21, 7, 3, 1; | [1], [3], [7], [21]; | 4 |
| 22 | 33, 3; | [1, 2], [11, 22]; | 2 |
| 23 | 23, 1; | [1], [23]; | 2 |
| 24 | 60; | [1, 2, 3, 4, 6, 8, 12, 24]; | 1 |
...
A conjectured relationship between a palindromic composition of sigma_0(n) = A000005(n) as n-th row of A384222 and the list of divisors of n as the n-th row of A027750 and a palindromic composition of sigma_1(n) = A000203(n) as the n-th row of A237270 and the diagram called "symmetric representation of sigma(n)" is as shown below with two examples.
.
For n = 10 the conjectured relationship is:
10th row of A384222.......................: [ 2 ], [ 2 ]
10th row of A027750.......................: 1, 2, 5, 10
10th row of A027750 with sublists.........: [ 1, 2 ], [ 5, 10]
10th row of A384149.......................: [ 3 ], [ 15 ]
10th row of this triangle.................: [ 15 ], [ 3 ]
10th row of the virtual sequence 2*A237270: [ 18 ], [ 18 ]
10th row of A237270.......................: [ 9 ], [ 9 ]
.
The symmetric representation of sigma_1(10) in the first quadrant is as follows:
.
_ _ _ _ _ _ 9
|_ _ _ _ _ |
| |_
|_ _|_
| |_ _ 9
|_ _ |
| |
| |
| |
| |
|_|
.
The diagram has two parts (or polygons) of areas [9], [9] respectively, so the 10th row of A237270 is [9], [9] and sigma_1(10) = A000203(10) = 18.
.
For n = 15 the conjectured relationship is:
15th row of A384222.......................: [ 1], [ 2 ], [ 1]
15th row of A027750.......................: 1, 3, 5, 15
15th row of A027750 with sublists.........: [ 1], [ 3, 5], [15]
15th row of A384149.......................: [ 1], [ 8 ], [15]
15th row of this triangle.................: [15], [ 8 ], [ 1]
15th row of the virtual sequence 2*A237270: [16], [ 16 ], [16]
15th row of A237270.......................: [ 8], [ 8 ], [ 8]
.
The symmetric representation of sigma_1(15) in the first quadrant is as follows:
.
_ _ _ _ _ _ _ _ 8
|_ _ _ _ _ _ _ _|
|
|_ _
|_ |_ 8
| |_
|_ _ |
|_|_ _ _ 8
| |
| |
| |
| |
| |
| |
| |
|_|
.
The diagram has three parts (or polygons) of areas [8], [8], [8] respectively, so the 15th row of A237270 is [8], [8], [8] and sigma_1(15) = A000203(15) = 24.
.
For the relationship with Dyck paths, partitions of n into consecutive parts and odd divisors of n see A237593, A235791, A237591 and A379630.
Cf.
A000105,
A174973,
A196020,
A236104,
A235791,
A237270,
A237271,
A237591,
A237593,
A240062,
A245092,
A262626,
A379288,
A379630,
A384222,
A384225,
A384226,
A384227.
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
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.
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
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.
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
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.
For the smallest parts see
A379632.
Cf.
A196020,
A212652,
A235791,
A236104,
A237048,
A237591,
A237593,
A245092,
A262626,
A379630,
A379634.
Showing 1-5 of 5 results.
Comments