A340847 a(n) is the number of vertices in the diagram of the symmetric representation of sigma(n) with subparts.

4, 6, 7, 10, 9, 13, 11, 14, 14, 15, 13, 23, 13, 17, 21, 22, 15, 26, 15, 25, 23, 21, 27, 35, 22, 21, 25, 29, 19, 41, 19, 30

Offset: 1

Author

Omar E. Pol, Jan 24 2021

Keywords

• nonn
• more

Comments

Theorem: Indices of even terms give A028982. Indices of odd terms give A028983.

If A001227(n) is odd then a(n) is even.

If A001227(n) is even then a(n) is odd.

The above sentences arise that the diagram is always symmetric for any value of n hence the number of edges is always an even number. Also from Euler's formula.

For another version see A340833 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        7         10           9              13
.
For n = 6 the diagram has 13 vertices so a(6) = 13.
On the other hand the diagram has 14 edges and two subparts or regions, so applying Euler's formula we have that a(6) = 14 - 2 + 1 = 13.
.
.                                                  _ _ _ _ _
.                            _ _ _ _ _            |_ _ _ _ _|
.        _ _ _ _            |_ _ _ _  |                     |_ _
.       |_ _ _ _|                   | |_                    |_  |
.               |_                  |_  |_ _                  |_|_ _
.                 |_ _                |_ _  |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   |_|                   |_|                     |_|
.
n:              7                    8                      9
a(n):          11                   14                     14
.
For n = 9 the diagram has 14 vertices so a(9) = 14.
On the other hand the diagram has 16 edges and three subparts or regions, so applying Euler's formula we have that a(9) = 16 - 3 + 1 = 14.
Another way for the illustration of initial terms is as follows:
--------------------------------------------------------------------------
.  n  a(n)                             Diagram
--------------------------------------------------------------------------
            _
   1   4   |_|  _
              _| |  _
   2   6     |_ _| | |  _
                _ _|_| | |  _
   3   7       |_ _|  _| | | |  _
                  _ _|  _| | | | |  _
   4  10         |_ _ _|  _|_| | | | |  _
                    _ _ _|  _ _| | | | | |  _
   5   9           |_ _ _| |  _ _| | | | | | |  _
                      _ _ _| |_|  _|_| | | | | | |  _
   6  13             |_ _ _ _|  _|  _ _| | | | | | | |  _
                        _ _ _ _|  _|  _ _| | | | | | | | |  _
   7  11               |_ _ _ _| |  _|  _ _|_| | | | | | | | |  _
                          _ _ _ _| |  _| |  _ _| | | | | | | | | |  _
   8  14                 |_ _ _ _ _| |_ _| |  _ _| | | | | | | | | | |  _
                            _ _ _ _ _|  _ _|_|  _ _|_| | | | | | | | | | |
   9  14                   |_ _ _ _ _| |  _|  _|  _ _ _| | | | | | | | | |
                              _ _ _ _ _| |  _|  _|  _ _ _| | | | | | | | |
  10  15                     |_ _ _ _ _ _| |  _|  _| |  _ _|_| | | | | | |
                                _ _ _ _ _ _| |  _|  _| |  _ _ _| | | | | |
  11  13                       |_ _ _ _ _ _| | |_ _|  _| |  _ _ _| | | | |
                                  _ _ _ _ _ _| |  _ _|  _|_|  _ _ _|_| | |
  12  23                         |_ _ _ _ _ _ _| |  _ _|  _ _| |  _ _ _| |
                                    _ _ _ _ _ _ _| |  _| |  _ _| |  _ _ _|
  13  13                           |_ _ _ _ _ _ _| | |  _| |_|  _| |
                                      _ _ _ _ _ _ _| | |_ _|  _|  _|
  14  17                             |_ _ _ _ _ _ _ _| |  _ _|  _|
                                        _ _ _ _ _ _ _ _| |  _ _|
  15  21                               |_ _ _ _ _ _ _ _| | |
                                          _ _ _ _ _ _ _ _| |
  16  22                                 |_ _ _ _ _ _ _ _ _|
...
		

Crossrefs

Cf. A001227 (number of subparts or regions).

Cf. A340848 (number of edges).

Cf. A340833 (numer of vertices in the diagram only with parts).

Cf. A317293 (total number of vertices in the unified diagram).

Cf. A000203, A028982, A028983, A060831, A196020, A236104, A235791, A237048, A237270, A237591, A237593, A239660, A245092, A262626, A279387, A280850, A280851, A296508, A335616, A340846.

Formula

a(n) = A340848(n) - A001227(n) + 1 (Euler's formula).