A340848 a(n) is the number of edges in the diagram of the symmetric representation of sigma(n) with subparts.

4, 6, 8, 10, 10, 14, 12, 14, 16, 16, 14, 24, 14, 18, 24, 22, 16, 28, 16, 26, 26, 22, 18, 36, 24, 22, 28, 30, 20, 44, 20, 30

Offset: 1

Author

Omar E. Pol, Jan 24 2021

Keywords

• nonn
• more

Comments

Since the diagram is symmetric so all terms are even numbers.

For another version see A340846 from which first differs at a(6).

For the definition of subparts see A279387. For more information about the subparts see also A237271, A280850, A280851, A296508, A335616.

Note that in this version of the diagram of the symmetric representation of sigma(n) all regions are called "subparts". The number of subparts equals A001227(n).

Examples

			Illustration of initial terms:
.                                                          _ _ _ _
.                                            _ _ _        |_ _ _  |_
.                                _ _ _      |_ _ _|             | |_|_
.                      _ _      |_ _  |_          |_ _          |_ _  |
.              _ _    |_ _|_        |_  |           | |             | |
.        _    |_  |       | |         | |           | |             | |
.       |_|     |_|       |_|         |_|           |_|             |_|
.
n:       1      2        3          4           5               6
a(n):    4      6        8         10          10              14
.
For n = 6 the diagram has 14 edges so a(6) = 14.
On the other hand the diagram has 13 vertices and two subparts or regions, so applying Euler's formula we have that a(6) = 13 + 2 - 1 = 14.
.                                                  _ _ _ _ _
.                            _ _ _ _ _            |_ _ _ _ _|
.        _ _ _ _            |_ _ _ _  |                     |_ _
.       |_ _ _ _|                   | |_                    |_  |
.               |_                  |_  |_ _                  |_|_ _
.                 |_ _                |_ _  |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   |_|                   |_|                     |_|
.
n:              7                    8                      9
a(n):          12                   14                     16
.
For n = 9 the diagram has 16 edges so a(9) = 16.
On the other hand the diagram has 14 vertices and three subparts or regions, so applying Euler's formula we have that a(9) = 14 + 3 - 1 = 16.
Another way for the illustration of initial terms is as follows:
--------------------------------------------------------------------------
.  n  a(n)                             Diagram
--------------------------------------------------------------------------
            _
   1   4   |_|  _
              _| |  _
   2   6     |_ _| | |  _
                _ _|_| | |  _
   3   8       |_ _|  _| | | |  _
                  _ _|  _| | | | |  _
   4  10         |_ _ _|  _|_| | | | |  _
                    _ _ _|  _ _| | | | | |  _
   5  10           |_ _ _| |  _ _| | | | | | |  _
                      _ _ _| |_|  _|_| | | | | | |  _
   6  14             |_ _ _ _|  _|  _ _| | | | | | | |  _
                        _ _ _ _|  _|  _ _| | | | | | | | |  _
   7  12               |_ _ _ _| |  _|  _ _|_| | | | | | | | |  _
                          _ _ _ _| |  _| |  _ _| | | | | | | | | |  _
   8  14                 |_ _ _ _ _| |_ _| |  _ _| | | | | | | | | | |  _
                            _ _ _ _ _|  _ _|_|  _ _|_| | | | | | | | | | |
   9  16                   |_ _ _ _ _| |  _|  _|  _ _ _| | | | | | | | | |
                              _ _ _ _ _| |  _|  _|  _ _ _| | | | | | | | |
  10  16                     |_ _ _ _ _ _| |  _|  _| |  _ _|_| | | | | | |
                                _ _ _ _ _ _| |  _|  _| |  _ _ _| | | | | |
  11  14                       |_ _ _ _ _ _| | |_ _|  _| |  _ _ _| | | | |
                                  _ _ _ _ _ _| |  _ _|  _|_|  _ _ _|_| | |
  12  24                         |_ _ _ _ _ _ _| |  _ _|  _ _| |  _ _ _| |
                                    _ _ _ _ _ _ _| |  _| |  _ _| |  _ _ _|
  13  14                           |_ _ _ _ _ _ _| | |  _| |_|  _| |
                                      _ _ _ _ _ _ _| | |_ _|  _|  _|
  14  18                             |_ _ _ _ _ _ _ _| |  _ _|  _|
                                        _ _ _ _ _ _ _ _| |  _ _|
  15  24                               |_ _ _ _ _ _ _ _| | |
                                          _ _ _ _ _ _ _ _| |
  16  22                                 |_ _ _ _ _ _ _ _ _|
...
		

Crossrefs

Cf. A001227 (number of subparts or regions).

Cf. A340847 (number of vertices).

Cf. A340846 (number of edges in the diagram only with parts).

Cf. A317292 (total number of edges in the unified diagram).

Cf. A000203, A060831, A196020, A236104, A235791, A237048, A237270, A237591, A237593, A239660, A245092, A262626, A279387, A280850, A280851, A296508, A335616, A340833.

Formula

a(n) = A340847(n) + A001227(n) - 1 (Euler's formula).