A280940 Irregular triangle read by rows: T(n,k) = number of subparts in the k-th part of the symmetric representation of sigma(n).

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, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Omar E. Pol, Jan 11 2017

Keywords

• nonn
• tabf
• more

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.

For more information about "subparts" see A279387, A279388 and A279391.

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.

Examples

			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

Row sums give A001227 (number of odd divisors of n).

Row lengths is A237271.

Cf. A000203, A196020, A235791, A236104, A237048, A237270, A237591, A237593, A244050, A245092, A249351, A250068, A262626, A279387, A279388, A279391.