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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Jan 23 2021

Keywords

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

Examples

			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                                 |_ _ _ _ _ _ _ _ _|
...

Links

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.

Programs

  • 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 *)

Formula

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

Extensions

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

A340846 a(n) is the number of edges in the diagram of the symmetric representation of sigma(n).

Original entry on oeis.org

4, 6, 8, 10, 10, 12, 12, 14, 16, 16, 14, 18, 14, 18, 22, 22, 16, 22, 16, 22, 26, 22, 18, 26, 24, 22, 28, 28, 20, 30, 20, 30, 30, 24, 28, 32, 22, 26, 32, 34, 22, 34, 22, 34, 38, 28, 24, 38, 32, 40, 34, 36, 24, 38, 38, 42, 36, 30, 26, 42, 26, 30, 46, 42, 40, 44, 28
Offset: 1

Views

Author

Omar E. Pol, Jan 24 2021

Keywords

Comments

Since the diagram is symmetric so all terms are even numbers.
For another version with subparts see A340848 from which first differs at a(6).

Examples

			Illustration of initial terms:
.                                                          _ _ _ _
.                                            _ _ _        |_ _ _  |_
.                                _ _ _      |_ _ _|             |   |_
.                      _ _      |_ _  |_          |_ _          |_ _  |
.              _ _    |_ _|_        |_  |           | |             | |
.        _    |_  |       | |         | |           | |             | |
.       |_|     |_|       |_|         |_|           |_|             |_|
.
n:       1      2        3          4           5               6
a(n):    4      6        8         10          10              12
.
For n = 6 the diagram has 12 edges so a(6) = 12.
On the other hand the diagram has 12 vertices 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):          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 parts 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  12             |_ _ _ _|  _|  _ _| | | | | | | |  _
                        _ _ _ _|  _|  _ _| | | | | | | | |  _
   7  12               |_ _ _ _| |  _|  _ _|_| | | | | | | | |  _
                          _ _ _ _| |  _| |  _ _| | | | | | | | | |  _
   8  14                 |_ _ _ _ _| |_ _| |  _ _| | | | | | | | | | |  _
                            _ _ _ _ _|  _ _|_|  _ _|_| | | | | | | | | | |
   9  16                   |_ _ _ _ _| |  _|  _|  _ _ _| | | | | | | | | |
                              _ _ _ _ _| |  _|  _|    _ _| | | | | | | | |
  10  16                     |_ _ _ _ _ _| |  _|     |  _ _|_| | | | | | |
                                _ _ _ _ _ _| |      _| |  _ _ _| | | | | |
  11  14                       |_ _ _ _ _ _| |  _ _|  _| |  _ _ _| | | | |
                                  _ _ _ _ _ _| |  _ _|  _|_|  _ _ _|_| | |
  12  18                         |_ _ _ _ _ _ _| |  _ _|  _ _| |  _ _ _| |
                                    _ _ _ _ _ _ _| |  _| |    _| |  _ _ _|
  13  14                           |_ _ _ _ _ _ _| | |  _|  _|  _| |
                                      _ _ _ _ _ _ _| | |_ _|  _|  _|
  14  18                             |_ _ _ _ _ _ _ _| |  _ _|  _|
                                        _ _ _ _ _ _ _ _| |  _ _|
  15  22                               |_ _ _ _ _ _ _ _| | |
                                          _ _ _ _ _ _ _ _| |
  16  22                                 |_ _ _ _ _ _ _ _ _|
...

Crossrefs

Cf. A237271 (number of parts or regions).
Cf. A340833 (number of vertices).
Cf. A340848 (number of edges in the diagram with subparts).
Cf. A317109 (total number of edges in the unified diagram).
Cf. A239931-A239934 (illustration of first 32 diagrams).
Cf. A000203, A005843, A196020, A236104, A235791, A237048, A237270, A237590, A237591, A237593, A239660, A245092, A262626, A340847.

Formula

a(n) = A340833(n) + A237271(n) - 1 (Euler's formula).

Extensions

More terms from Omar E. Pol, Oct 28 2021

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

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Jan 24 2021

Keywords

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).

A348406 Number of vertices on the axis of symmetry of the symmetric representation of sigma(n) with subparts.

Original entry on oeis.org

2, 2, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 1, 3, 2, 1, 2, 1, 3, 1, 1, 1, 3, 2, 1, 1, 3, 1, 3, 1, 2, 1, 1, 3, 2, 1, 1, 1, 3, 1, 3, 1, 1, 3, 1, 1, 3, 2, 2, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 3, 2, 1, 3, 1, 1, 1, 3, 1, 4, 1, 1, 1, 1, 3, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 3, 1, 3, 3, 1, 1, 1, 1, 3, 1, 2, 3, 2, 1, 1, 1, 3, 1
Offset: 1

Views

Author

Omar E. Pol, Oct 17 2021

Keywords

Comments

The number of middle divisors of n is equal to a(n) - 1.
For the definition of "subparts" see A279387.

Examples

			For n = 2, 6 and 10 the symmetric representation of sigma(n) with subparts respectively looks like this:
.
.           _       _       _
.         _| |     | |     | |
.    2   |_ _|     | |     | |
.               _ _| |     | |
.              |  _ _|     | |
.         _ _ _| |_|    _ _| |
.    6   |_ _ _ _|     |  _ _|
.                   _ _|_|
.                  |  _|
.         _ _ _ _ _| |
.   10   |_ _ _ _ _ _|
.
For n = 2 there are two vertices on the axis of symmetry hence the number of middle divisors of 2 is equal to 2 - 1 = 1.
For n = 6 there are three vertices on the axis of symmetry hence the number of middle divisors of 6 is equal to 3 - 1 = 2.
For n = 10 there is only one vertex on the axis of symmetry hence the number of middle divisors of 10 is equal to 1 - 1 = 0.

Links

Crossrefs

Companion of A348364.
Cf. A067742, A071090, A071540, A071562, A071563, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A240542, A279387, A281007, A299761, A303297, A340833, A340847, A346868, A347950, A348327.

Programs

  • Mathematica
    a[n_] := 1 + DivisorSum[n, 1 &, n/2 <= #^2 < 2*n &]; Array[a, 100] (* Amiram Eldar, Oct 17 2021 *)
  • PARI
    A067742(n) = sumdiv(n, d, my(d2 = d^2); n / 2 < d2 && d2 <= n << 1); \\ From A067742
A348406(n) = (1 + A067742(n));

Formula

a(n) = 1 + A067742(n).
Showing 1-4 of 4 results.

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