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.
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
Examples
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].
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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|
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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.
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