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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.

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A279029 Numbers k with the property that the smallest and the largest Dyck path of the symmetric representation of sigma(k) do not share line segments.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 136, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252, 256
Offset: 1

Views

Author

Omar E. Pol, Dec 08 2016

Keywords

Comments

Numbers k such that the symmetric representation of sigma(k) is formed by only one part, or that it's formed by only two parts and they meet at the center.
Numbers k whose total length of all line segments of the symmetric representation of sigma(k) is equal to 4*k (cf. A348705). For the positive integers k that are not in this sequence the mentioned total length is < 4*k. - Omar E. Pol, Nov 02 2021
From Hartmut F. W. Hoft, Jan 25 2025: (Start)
The following three statements are equivalent for numbers k >= 1:
(1) The symmetric representation of sigma(k) is formed of 2 parts that meet at the diagonal.
(2) A249223(k, A003056(k)) = 0 is the only 0 in row k of the triangle, and A237591(k, A003056(k)) = 1.
(3) Row k of the triangle in A341969 contains a single 0 at the center position.
The following two statements are equivalent for numbers k >= 1:
(1) The symmetric representation of sigma(k) consists of a single part.
(2) Row k of the triangle in A249223 contains no 0. (End)
This sequence is the disjoint union of A262259 and A174973. Each member of A262259 has the form k = q*(2*q + 1) where 2*q + 1 is prime; also A003056(k) = 2*q. Therefore [q, 2*q] contains a divisor q of k while (q, 2*q] contains no divisor of k. A262259 is a subsequence of A298259, see also A240542. - Hartmut F. W. Hoft, Mar 24 2025
My two links below give detailed proofs for the last comment. - Hartmut F. W. Hoft, Jun 10 2025

Examples

			1, 2, 3, 4, 6, 8, 10, 12 and 16 are in the sequence because the smallest and the largest Dyck path of their symmetric representation of sigma do not share line segments, as shown below.
Illustration of initial terms:
  n
  .      _ _ _ _   _   _   _   _       _
  1     |_| | | | | | | | | | | |     | |
  2     |_ _|_| | | | | | | | | |     | |
  3     |_ _|  _|_| | | | | | | |     | |
  4     |_ _ _|    _|_| | | | | |     | |
         _ _ _|  _|  _ _|_| | | |     | |
  6     |_ _ _ _|  _| |  _ _|_| |     | |
         _ _ _ _| |_ _|_|    _ _|     | |
  8     |_ _ _ _ _|  _|     |    _ _ _| |
         _ _ _ _ _| |      _|   |  _ _ _|
  10    |_ _ _ _ _ _|  _ _|    _| |
         _ _ _ _ _ _| |      _|  _|
  12    |_ _ _ _ _ _ _|  _ _|  _|
                        |  _ _|
                        | |
         _ _ _ _ _ _ _ _| |
  16    |_ _ _ _ _ _ _ _ _|
  ...

Links

Crossrefs

UNION of A174973 and A262259.
Positions of 0's in A279228.
Complement is A279244.
Cf. A000203, A003056, A008586, A174973, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A249223, A262259, A323648, A348705, A379968 (characteristic function).
Cf. A240542, A298856.

Programs

  • Mathematica
    (* Function path[] is defined in A237270 *)
a279029Q[n_] := Length[Select[Transpose[{Take[path[n], {2,-2}], path[n-1]}], #[[1]]==#[[2]]&]]<=1
a279029[n_] := Select[Range[n], a279029Q]
a279029[256]
(* Alternate, faster function based on A249223 *)
a003056[n_] := Floor[(Sqrt[8n+1]-1)/2]
t249223[n_] :=FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, a003056[n]]]
a262259Q[n_] := Position[t249223[n], 0]=={{a003056[n]}}&&Last[t237591[n]]==1
a174973Q[n_] := !MemberQ[t249223[n], 0]
a279029[n_] := Select[Range[n], a262259Q[#]||a174973Q[#]&]
a279029[256] (* Hartmut F. W. Hoft, Jan 25 2025 *)
  • PARI
    is_A279029 = A379968; \\ Antti Karttunen, Jan 12 2025

Formula

a(n) = A323648(n-1) + 1, for n >= 2. - Hartmut F. W. Hoft, Jan 25 2025

A279228 Number of unit steps that are shared by the smallest and largest Dyck path of the symmetric representation of sigma(n).

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 4, 0, 2, 0, 8, 0, 10, 2, 4, 0, 14, 0, 16, 0, 6, 6, 20, 0, 16, 8, 10, 0, 26, 0, 28, 0
Offset: 1

Views

Author

Omar E. Pol, Dec 08 2016

Keywords

Comments

a(n) is also the number of unit steps that are shared by the largest Dyck path of the symmetric representation of sigma(n) and the largest Dyck path of the symmetric representation of sigma(n-1), in a quadrant of the square grid.
For more information about the Dyck paths of the symmetric representation of sigma(n) see A237593.

Examples

			Illustration of initial terms (n = 1..12) using the spiral described in A239660:
.               _ _ _ _ _ _
.              |  _ _ _ _ _|_ _ _ _ _
.         0   _| |         |_ _ _ _ _|
.           _|_ _|                   |_ _ 2
.       _ _| |      _ _ _ _          |_  |
.      |  _ _|  0 _|  _ _ _|_ _ _      |_|_ _
.      | |      _|   |     |_ _ _|  2      | |
.      | |     |  _ _|           |_ _      | |
.      | |     | |    0 _ _        | |     | |
.      | |     | |     |  _|_ 0    | |     | |
.     _|_|    _|_|    _|_| |_|    _|_|    _|_|    _
.    | |     | |     | |         | |     | |     | |
.    | |     | |     |_|_ _     _| |     | |     | |
.    | |     | |      0|_ _|_ _|  _|     | |     | |
.    | |     |_|_          |_ _ _|0   _ _| |     | |
.    | |         |_                 _|  _ _|     | |
.    |_|_ _     4  |_ _ _ _        |  _|    _ _ _| |
.          |_      |_ _ _ _|_ _ _ _| |  0 _|    _ _|
.            |_            |_ _ _ _ _|  _|     |
.         8    |                       |      _|
.              |_ _ _ _ _ _            |  _ _|
.              |_ _ _ _ _ _|_ _ _ _ _ _| |      0
.                          |_ _ _ _ _ _ _|
.
.
For an illustration of the following examples see the last lap of the above spiral starting in the first quadrant.
For n = 9 the Dyck paths of the symmetric representation of sigma(9) share 2 unit steps, so a(9) = 2.
For n = 10 the Dyck paths of the symmetric representation of sigma(10) meet at the center, but they do not share unit steps, so a(10) = 0.
For n = 11 the Dyck paths of the symmetric representation of sigma(11) share 8 unit steps, so a(11) = 8.
For n = 12 the Dyck paths of the symmetric representation of sigma(12) do not share unit steps, so a(12) = 0.
Note that we can find the spiral on the terraces of the stepped pyramid described in A244050.

Crossrefs

Cf. A279029 gives the indices of the zero values.
Cf. A279244 gives the indices of the positive values.
Cf. A000203, A005843, A008586, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A244050, A244361, A244363, A245092, A262626, A348705.

Formula

a(n) = 2*n - A244363(n) = 2*(n - A244361(n)).
a(n) = A008586(n) - A348705(n). - Omar E. Pol, Dec 13 2021

A323648 Numbers k such that the largest Dyck path of the symmetric representation of sigma(k) does not share any line segment with the largest Dyck path of the symmetric representation of sigma(k+1).

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 15, 17, 19, 23, 27, 29, 31, 35, 39, 41, 47, 53, 55, 59, 63, 65, 71, 77, 79, 83, 87, 89, 95, 99, 103, 107, 111, 119, 125, 127, 131, 135, 139, 143, 149, 155, 159, 161, 167, 175, 179, 191, 195, 197, 199, 203, 207, 209, 215, 219, 223, 227, 233, 239, 251, 255
Offset: 1

Views

Author

Omar E. Pol, Apr 02 2019

Keywords

Comments

Equivalently, numbers k such that in the perspective view of the stepped pyramid described in A245092, the steps of the n-th level do not share any vertical face with the steps of the level n + 1, starting from the top of the pyramid.
a(2) = 2 is the only even number in the sequence.
For more information about the Dyck paths, the connection with the sum of divisors function A000203, and the connection with the theory of partitions see A237593.

Crossrefs

Cf. A174973, A000203, A196020, A236104, A237270, A237271, A237591, A237593, A244145, A245092, A245192, A262259, A262626, A279029, A279244.

Programs

  • Mathematica
    (* Function path[] is defined in A237270 *)
a323648Q[n_] := Length[Select[Transpose[{Take[path[n+1], {2,-2}], path[n]}], #[[1]]==#[[2]]&]]<=1
a323648[n_] := Select[Range[n], a323648Q]
a323648[255]
(* Functions a262259Q[ ] and a174973Q[ ] are defined in A279029 *)
a323648[n_] := Select[Range[n], a262259Q[#+1]||a174973Q[#+1]&]
a323648[255] (* Hartmut F. W. Hoft, Jan 25 2025 *)

Formula

a(n) = A279029(n+1) - 1, for n >= 1. - Hartmut F. W. Hoft, Jan 25 2025

Extensions

a(17)-a(63) by Hartmut F. W. Hoft, Jan 25 2025
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