A340833 a(n) is the number of vertices in the diagram of the symmetric representation of sigma(n).

4, 6, 7, 10, 9, 12, 11, 14, 14, 15, 13, 18, 13, 17, 20, 22, 15, 22, 15, 22, 23, 21, 17, 26, 22, 21, 25, 28, 19, 30, 19, 30, 27, 23, 26, 32, 21, 25, 29, 34, 21, 34, 21, 33, 36, 27, 23, 38, 30, 38, 31, 35, 23, 38, 35, 42, 33, 29, 25, 42, 25, 29, 42, 42, 37, 44, 27

Offset

1,1

Comments

If A237271(n) is odd then a(n) is even.

If A237271(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.

Indices of odd terms give A071561.

Indices of even terms give A071562.

For another version with subparts see A340847 from which first differs at a(6).

The parity of this sequence is also the characteristic function of numbers that have no middle divisors (cf. A348327). - Omar E. Pol, Oct 14 2021

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log-log scatterplot of a(n) for n=1..10^4, accentuating a(m) for m=1..2^8 for clarity, and labeling a(k) for k=1..24.

Formula

a(n) = A340846(n) - A237271(n) + 1 (Euler's formula).

Example

Illustration of initial terms:
.                                                          _ _ _ _
.                                            _ _ _        |_ _ _  |_
.                                _ _ _      |_ _ _|             |   |_
.                      _ _      |_ _  |_          |_ _          |_ _  |
.              _ _    |_ _|_        |_  |           | |             | |
.        _    |_  |       | |         | |           | |             | |
.       |_|     |_|       |_|         |_|           |_|             |_|
.
n:       1      2        3          4           5               6
a(n):    4      6        7         10           9              12
.
For n = 6 the diagram has 12 vertices so a(6) = 12.
On the other hand the diagram has 12 edges and only one part or region, so applying Euler's formula we have that a(6) = 12 - 1 + 1 = 12.
.                                                  _ _ _ _ _
.                            _ _ _ _ _            |_ _ _ _ _|
.        _ _ _ _            |_ _ _ _  |                     |_ _
.       |_ _ _ _|                   | |_                    |_  |
.               |_                  |_  |_ _                  |_|_ _
.                 |_ _                |_ _  |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   |_|                   |_|                     |_|
.
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 parts 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  12             |_ _ _ _|  _|  _ _| | | | | | | |  _
                        _ _ _ _|  _|  _ _| | | | | | | | |  _
   7  11               |_ _ _ _| |  _|  _ _|_| | | | | | | | |  _
                          _ _ _ _| |  _| |  _ _| | | | | | | | | |  _
   8  14                 |_ _ _ _ _| |_ _| |  _ _| | | | | | | | | | |  _
                            _ _ _ _ _|  _ _|_|  _ _|_| | | | | | | | | | |
   9  14                   |_ _ _ _ _| |  _|  _|  _ _ _| | | | | | | | | |
                              _ _ _ _ _| |  _|  _|    _ _| | | | | | | | |
  10  15                     |_ _ _ _ _ _| |  _|     |  _ _|_| | | | | | |
                                _ _ _ _ _ _| |      _| |  _ _ _| | | | | |
  11  13                       |_ _ _ _ _ _| |  _ _|  _| |  _ _ _| | | | |
                                  _ _ _ _ _ _| |  _ _|  _|_|  _ _ _|_| | |
  12  18                         |_ _ _ _ _ _ _| |  _ _|  _ _| |  _ _ _| |
                                    _ _ _ _ _ _ _| |  _| |    _| |  _ _ _|
  13  13                           |_ _ _ _ _ _ _| | |  _|  _|  _| |
                                      _ _ _ _ _ _ _| | |_ _|  _|  _|
  14  17                             |_ _ _ _ _ _ _ _| |  _ _|  _|
                                        _ _ _ _ _ _ _ _| |  _ _|
  15  20                               |_ _ _ _ _ _ _ _| | |
                                          _ _ _ _ _ _ _ _| |
  16  22                                 |_ _ _ _ _ _ _ _ _|
...

Mathematica

MapAt[# + 1 &, #, 1] &@ Map[Length@ Union[Join @@ #] - 1 &, Partition[Prepend[#, {{0, 0}}], 2, 1]] &@ Table[{{0, 0}}~Join~Accumulate[Join[#, Reverse[Reverse /@ (-1*#)]]] &@ MapIndexed[Which[#2 == 1, {#1, 0}, Mod[#2, 2] == 0, {0, #1}, True, {-#1, 0}] & @@ {#1, First[#2]} &, If[Length[#] == 0, {n, n}, Join[{n}, #, {n - Total[#]}]]] &@ Differences[n - Array[(Ceiling[(n + 1)/# - (# + 1)/2]) &, Floor[(Sqrt[8 n + 1] - 1)/2]]], {n, 67}] (* Michael De Vlieger, Oct 27 2021 *)

Crossrefs

Parity gives A348327.

Cf. A237271 (number of parts or regions).

Cf. A340846 (number of edges).

Cf. A340847 (number of vertices in the diagram with subparts).

Cf. A294723 (total number of vertices in the unified diagram).

Cf. A239931-A239934 (illustration of first 32 diagrams).

Cf. A000203, A071561, A071562, A196020, A236104, A235791, A237048, A237270, A237590, A237591, A237593, A239660, A245092, A262626, A340848.

Sequence in context: A087790 A278960 A344570 * A340847 A286494 A086170

Adjacent sequences: A340830 A340831 A340832 * A340834 A340835 A340836

Keyword

nonn,look

Author

Omar E. Pol, Jan 23 2021

Extensions

Terms a(33) and beyond from Michael De Vlieger, Oct 27 2021

Status

approved