OFFSET
1,8
COMMENTS
The "subparts" of the symmetric representation of sigma(n) are the regions that arise after the dissection of the symmetric representation of sigma(n) into successive layers of width 1.
The number of subparts in the symmetric representation of sigma(n) equals the number of odd divisors of n.
Note that we can find the symmetric representation of sigma(n) as the terraces at the n-th level (starting from the top) of the stepped pyramid described in A245092.
EXAMPLE
Triangle begins (n = 1..21):
1;
1;
1, 1;
1;
1, 1;
2;
1, 1;
1;
1, 1, 1;
1, 1;
1, 1;
2;
1, 1;
1, 1;
1, 2, 1;
1;
1, 1;
3;
1, 1;
2;
1, 1, 1, 1;
...
For n = 12 we have that the 11th row of triangle A237593 is [6, 3, 1, 1, 1, 1, 3, 6] and the 12th row of the same triangle is [7, 2, 2, 1, 1, 2, 2, 7], so the diagram of the symmetric representation of sigma(12) = 28 is constructed as shown below in Figure 1:
. _ _
. | | | |
. | | | |
. | | | |
. | | | |
. | | | |
. _ _ _| | _ _ _| |
. 28 _| _ _| 23 _| _ _ _|
. _| | _| _| |
. | _| | _| _|
. | _ _| | |_ _|
. _ _ _ _ _ _| | _ _ _ _ _ _| | 5
. |_ _ _ _ _ _ _| |_ _ _ _ _ _ _|
.
. Figure 1. The symmetric Figure 2. After the dissection
. representation of sigma(12) of the symmetric representation
. has only one part which of sigma(12) into layers of
. contains 28 cells, so width 1 we can see two "subparts"
. A237271(12) = 1, and that contain 23 and 5 cells
. A000203(12) = 28. respectively, so the 12th row of
. this triangle is [2], and the
. row sum is A001227(12) = 2, equaling
. the number of odd divisors of 12.
.
For n = 15 we have that the 14th row of triangle A237593 is [8, 3, 1, 2, 2, 1, 3, 8] and the 15th row of the same triangle is [8, 3, 2, 1, 1, 1, 1, 2, 3, 8], so the diagram of the symmetric representation of sigma(15) = 24 is constructed as shown below in Figure 3:
. _ _
. | | | |
. | | | |
. | | | |
. | | | |
. 8 | | 8 | |
. | | | |
. | | | |
. _ _ _|_| _ _ _|_|
. 8 _ _| | 7 _ _| |
. | _| | _ _|
. _| _| _| |_|
. |_ _| |_ _| 1
. 8 | 8 |
. _ _ _ _ _ _ _ _| _ _ _ _ _ _ _ _|
. |_ _ _ _ _ _ _ _| |_ _ _ _ _ _ _ _|
.
. Figure 3. The symmetric Figure 4. After the dissection
. representation of sigma(15) of the symmetric representation
. has three parts of size 8 of sigma(15) into layers of
. because every part contains width 1 we can see four "subparts".
. 8 cells, so A237271(15) = 3, The first and third part contains
. and A000203(15) = 8+8+8 = 24. one subpart each. The second part contains
. two subparts, so the 15th row of this
. triangle is [1, 2, 1], and the row
. sum is A001227(15) = 4, equaling the
. number of odd divisors of 15.
.
CROSSREFS
KEYWORD
nonn,tabf,more
AUTHOR
Omar E. Pol, Jan 11 2017
STATUS
approved