cp's OEIS Frontend

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.

A317303 Numbers k such that both Dyck paths of the symmetric representation of sigma(k) have a central peak.

Original entry on oeis.org

2, 7, 8, 9, 16, 17, 18, 19, 20, 29, 30, 31, 32, 33, 34, 35, 46, 47, 48, 49, 50, 51, 52, 53, 54, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 154, 155, 156, 157, 158, 159, 160
Offset: 1

Views

Author

Omar E. Pol, Aug 27 2018

Keywords

Comments

Also triangle read by rows which gives the odd-indexed rows of triangle A014132.
There are no triangular number (A000217) in this sequence.
For more information about the symmetric representation of sigma see A237593 and its related sequences.
Equivalently, numbers k with the property that both Dyck paths of the symmetric representation of sigma(k) have an odd number of peaks. - Omar E. Pol, Sep 13 2018

Examples

			Written as an irregular triangle in which the row lengths are the odd numbers, the sequence begins:
    2;
    7,   8,   9;
   16,  17,  18,  19,  20;
   29,  30,  31,  32,  33,  34,  35;
   46,  47,  48,  49,  50,  51,  52,  53,  54;
   67,  68,  69,  70,  71,  72,  73,  74,  75,  76,  77;
   92,  93,  94,  95,  96,  97,  98,  99, 100, 101, 102, 103, 104;
  121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135;
...
Illustration of initial terms:
-----------------------------------------------------------
   k   sigma(k)   Diagram of the symmetry of sigma
-----------------------------------------------------------
                    _         _ _ _             _ _ _ _ _
                  _| |       | | | |           | | | | | |
   2      3      |_ _|       | | | |           | | | | | |
                             | | | |           | | | | | |
                            _|_| | |           | | | | | |
                          _|  _ _|_|           | | | | | |
                  _ _ _ _|  _| |               | | | | | |
   7      8      |_ _ _ _| |_ _|               | | | | | |
   8     15      |_ _ _ _ _|              _ _ _| | | | | |
   9     13      |_ _ _ _ _|             |  _ _ _|_| | | |
                                        _| |    _ _ _|_| |
                                      _|  _|   |  _ _ _ _|
                                  _ _|  _|  _ _| |
                                 |  _ _|  _|    _|
                                 | |     |     |
                  _ _ _ _ _ _ _ _| |  _ _|  _ _|
  16     31      |_ _ _ _ _ _ _ _ _| |  _ _|
  17     18      |_ _ _ _ _ _ _ _ _| | |
  18     39      |_ _ _ _ _ _ _ _ _ _| |
  19     20      |_ _ _ _ _ _ _ _ _ _| |
  20     42      |_ _ _ _ _ _ _ _ _ _ _|
.
For the first nine terms of the sequence we can see in the above diagram that both Dyck path (the smallest and the largest) of the symmetric representation of sigma(k) have a central peak.
Compare with A317304.

Crossrefs

Column 1 gives A130883, n >= 1.
Column 2 gives A033816, n >= 1.
Row sums give the odd-indexed terms of A006002.
Right border gives the positive terms of A014107, also the odd-indexed terms of A000096.
The union of A000217, A317304 and this sequence gives A001477.
Some other sequences related to the central peak or the central valley of the symmetric representation of sigma are A000217, A000384, A007606, A007607, A014105, A014132, A162917, A161983, A317304. See also A317306.
Cf. A000203, A005408, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A249351, A262626.

A317304 Numbers k with the property that both Dyck paths of the symmetric representation of sigma(k) have a central valley.

Original entry on oeis.org

4, 5, 11, 12, 13, 14, 22, 23, 24, 25, 26, 27, 37, 38, 39, 40, 41, 42, 43, 44, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149
Offset: 1

Views

Author

Omar E. Pol, Aug 27 2018

Keywords

Comments

Also triangle read by rows which gives the even-indexed rows of triangle A014132.
There are no triangular number (A000217) in this sequence.
For more information about the symmetric representation of sigma see A237593 and its related sequences.
Equivalently, numbers k with the property that both Dyck paths of the symmetric representation of sigma(k) have an even number of peaks. - Omar E. Pol, Sep 13 2018

Examples

			Written as an irregular triangle in which the row lengths are the positive even numbers, the sequence begins:
    4,   5;
   11,  12,  13,  14;
   22,  23,  24,  25,  26,  27;
   37,  38,  39,  40,  41,  42,  43,  44;
   56,  57,  58,  59,  60,  61,  62,  63,  64,  65;
   79,  80,  81,  82,  83,  84,  85,  86,  87,  88,  89,  90;
  106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119;
...
Illustration of initial terms:
-------------------------------------------------
   k  sigma(k)  Diagram of the symmetry of sigma
-------------------------------------------------
                       _ _           _ _ _ _
                      | | |         | | | | |
                     _| | |         | | | | |
                 _ _|  _|_|         | | | | |
   4      7     |_ _ _|             | | | | |
   5      6     |_ _ _|             | | | | |
                                 _ _|_| | | |
                               _|    _ _|_| |
                             _|     |  _ _ _|
                            |      _|_|
                 _ _ _ _ _ _|  _ _|
  11     12     |_ _ _ _ _ _| |  _|
  12     28     |_ _ _ _ _ _ _| |
  13     14     |_ _ _ _ _ _ _| |
  14     24     |_ _ _ _ _ _ _ _|
.
For the first six terms of the sequence we can see in the above diagram that both Dyck path (the smallest and the largest) of the symmetric representation of sigma(k) have a central valley.
Compare with A317303.

Crossrefs

Row sums give A084367. n >= 1.
Column 1 gives A084849, n >= 1.
Column 2 gives A096376, n >= 1.
Right border gives the nonzero terms of A014106.
The union of A000217, A317303 and this sequence gives A001477.
Some other sequences related to the central peak or the central valley of the symmetric representation of sigma are A000217, A000384, A007606, A007607, A014105, A014132, A162917, A161983, A317303. See also A317306.
Cf. A000203, A005843, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A249351, A262626.

A317307 Sum of divisors of powers of 2 and sum of divisors of even perfect numbers.

Original entry on oeis.org

1, 3, 7, 12, 15, 31, 56, 63, 127, 255, 511, 992, 1023, 2047, 4095, 8191, 16256, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67100672, 67108863, 134217727, 268435455, 536870911, 1073741823, 2147483647, 4294967295, 8589934591, 17179738112, 17179869183
Offset: 1

Views

Author

Omar E. Pol, Aug 25 2018

Keywords

Comments

Sum of divisors of the numbers k such that the symmetric representation of sigma(k) has only one part, and apart from the central width, the rest of the widths are 1's.
Note that the above definition implies that the central width of the symmetric representation of sigma(k) is 1 or 2. For powers of 2 the central width is 1. For even perfect numbers the central width is 2 (see example).

Examples

			Illustration of initial terms. a(n) is the area (or the number of cells) of the n-th region of the diagram:
.        _ _   _   _   _               _                       _       _
.   1   |_| | | | | | | |             | |                     | |     | |
.   3   |_ _|_| | | | | |             | |                     | |     | |
.        _ _|  _|_| | | |             | |                     | |     | |
.   7   |_ _ _|    _|_| |             | |                     | |     | |
.        _ _ _|  _|  _ _|             | |                     | |     | |
.  12   |_ _ _ _|  _|                 | |                     | |     | |
.        _ _ _ _| |                   | |                     | |     | |
.  15   |_ _ _ _ _|              _ _ _| |                     | |     | |
.                               |  _ _ _|                     | |     | |
.                              _| |                           | |     | |
.                            _|  _|                           | |     | |
.                        _ _|  _|                             | |     | |
.                       |  _ _|                               | |     | |
.                       | |                          _ _ _ _ _| |     | |
.        _ _ _ _ _ _ _ _| |                         |  _ _ _ _ _|     | |
.  31   |_ _ _ _ _ _ _ _ _|                         | |    _ _ _ _ _ _| |
.                                                _ _| |   |  _ _ _ _ _ _|
.                                            _ _|  _ _|   | |
.                                           |    _|    _ _| |
.                                          _|  _|     |  _ _|
.                                         |  _|      _| |
.                                    _ _ _| |      _|  _|
.                                   |  _ _ _|  _ _|  _|
.                                   | |       |  _ _|
.                                   | |  _ _ _| |
.                                   | | |  _ _ _|
.        _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | |
.   56  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.                                       | |
.                                       | |
.        _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.   63  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
The diagram shows the first eight terms of the sequence. The symmetric representation of sigma of the numbers A317306: 1, 2, 4, 6, 8, 16, 28, 32, ..., has only one part, and apart from the central width, the rest of the widths are 1's.

Crossrefs

Union of nonzero terms of A000225 and A139256.
Odd terms give the nonzeros terms of A000225.
Even terms give A139256.
Subsequence of A317305.
Cf. A249351 (the widths).
Cf. A000203, A000396, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A262626, A317306.

Programs

  • Mathematica
    DivisorSigma[1, #] &@ Union[2^Range[0, Floor@ Log2@ Last@ #], #] &@ Array[2^(# - 1) (2^# - 1) &@ MersennePrimeExponent@ # &, 7] (* Michael De Vlieger, Aug 25 2018, after Robert G. Wilson v at A000396 *)

Formula

a(n) = A000203(A317306(n)).

A368582 a(n) = floor((sigma(n) + 1) / 2).

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 4, 8, 7, 9, 6, 14, 7, 12, 12, 16, 9, 20, 10, 21, 16, 18, 12, 30, 16, 21, 20, 28, 15, 36, 16, 32, 24, 27, 24, 46, 19, 30, 28, 45, 21, 48, 22, 42, 39, 36, 24, 62, 29, 47, 36, 49, 27, 60, 36, 60, 40, 45, 30, 84, 31, 48, 52, 64, 42, 72, 34, 63
Offset: 1

Views

Author

Peter Luschny, Dec 31 2023

Keywords

Crossrefs

Cf. A000203, A000079 (2^n), A000396 (perfect), A088580, A317306, A368207 (Bacher).

Programs

  • Julia
    using Nemo
A368582(n::Int) = div(divisor_sigma(n, 1) + 1, 2)
println([A368582(n) for n in 1:68])
  • Mathematica
    Array[Floor[(DivisorSigma[1, #] + 1)/2] &, 120] (* Michael De Vlieger, Dec 31 2023 *)
  • PARI
    a(n) = (sigma(n)+1)\2; \\ Michel Marcus, Jan 03 2024

Formula

a(p) = (p + 1) / 2 for all odd prime p.
a(n) = n <=> n term of union of A000079 and A000396. (If there are no odd perfect numbers also of A317306).
a(n) = floor(A088580(n)/2). - Omar E. Pol, Dec 31 2023
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.