A348705 -id:A348705 - cp's OEIS frontend

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.

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

Omar E. Pol, Dec 08 2016

nonn

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

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

Antti Karttunen, Table of n, a(n) for n = 1..20000
Hartmut F. W. Hoft, Proof: Dyck Paths of SRS(n) meeting only at diagonal
Hartmut F. W. Hoft, Proofs for A279029 = A174973 union A262259

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.

(* 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 *)

is_A279029 = A379968; \\ Antti Karttunen, Jan 12 2025

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

Omar E. Pol, Dec 08 2016

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.

			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.

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.

a(n) = 2*n - A244363(n) = 2*(n - A244361(n)).

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

A279244 Numbers k with the property that both the smallest and the largest Dyck path of the symmetric representation of sigma(k) share some line segments.

Original entry on oeis.org

5, 7, 9, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92

Offset: 1

Omar E. Pol, Dec 08 2016

nonn

Numbers k such that the symmetric representation of sigma(k) is formed by more than two parts, or that it is formed by only two parts and they do not meet at the center.
Numbers k whose total length of all line segments of the symmetric representation of sigma(k) is < 4*k (cf. A348705). - Omar E. Pol, Nov 02 2021
a(n) is also 1 plus the n-th term of the complement of A323648. - Hartmut F. W. Hoft, Feb 22 2025

			5, 7, 9, 11, 13, 14, and 15 are in the sequence because the smallest and the largest Dyck path of their symmetric representation of sigma share some line segments, as shown below.
Illustration of initial terms:
n
.              _   _   _   _   _ _ _
.             | | | | | | | | | | | |
.             | | | | | | | | | | | |
.            _|_| | | | | | | | | | |
.      _ _ _|    _|_| | | | | | | | |
5     |_ _ _|  _|  _ _|_| | | | | | |
.      _ _ _ _|  _| |  _ _|_| | | | |
7     |_ _ _ _| |_ _|_|    _ _|_| | |
.      _ _ _ _ _|  _|     |  _ _ _|_|
9     |_ _ _ _ _| |      _|_| |
.      _ _ _ _ _ _|  _ _|    _|
11    |_ _ _ _ _ _| |  _|  _|
.      _ _ _ _ _ _ _| |_ _|
13    |_ _ _ _ _ _ _| |
14    |_ _ _ _ _ _ _ _|
15    |_ _ _ _ _ _ _ _|
...

Complement of A279029.
Indices of positive terms in A279228.
Subsequence of A238524.
Cf. A000203, A008586, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A279029, A279244, A323648, A348705.

(* Function a279029Q is defined in A279029 *)
a279244[n_] := Select[Range[n], !a279029Q[#]&]
a279244[92] (* Hartmut F. W. Hoft, Feb 20 2025 *)

A348854 a(n) is the total length of all line segments in an octant of the symmetric representation of sigma(n).

Original entry on oeis.org

2, 4, 6, 8, 9, 12, 12, 16, 17, 20, 18, 24, 21, 27, 28, 32, 27, 36, 30, 40, 39, 41, 36, 48, 42, 48, 49, 56, 45, 60, 48, 64

Offset: 1

Omar E. Pol, Nov 01 2021

One half of the total length of all line segments of the symmetric representation of sigma(n).

If the symmetric representation of sigma(n) has only one part (cf. A174973) or if it has two parts and they meet at the center of the Dyck path (cf. A262259) then a(n) = 2*n, otherwise a(n) < 2*n. In other words: if n is a term of A279029 then a(n) = 2*n, otherwise a(n) < 2*n.

Cf. A005843 (upper bounds).
For illustrations see A348705.
Cf. A174973, A235791, A236104, A237270, A237271, A237591, A237593, A238443, A239660, A239931-A239934, A245092, A262259, A279029.

a(n) = A348705(n)/2.

A362817 Irregular triangle read by rows: T(n,k) (n>=1, k>=1) is the number of edges of the k-th polygon (or part), from left to right, of the symmetric representation of sigma(n).

Original entry on oeis.org

4, 6, 4, 4, 10, 4, 4, 12, 4, 4, 14, 4, 6, 4, 8, 8, 4, 4, 18, 4, 4, 8, 8, 4, 12, 4, 22, 4, 4, 22, 4, 4, 22, 4, 8, 8, 4, 8, 8, 4, 4, 26, 4, 10, 4, 8, 8, 4, 8, 8, 4, 28, 4, 4, 30, 4, 4, 30

Offset: 1

Omar E. Pol, May 04 2023

Row n is [4, 4] if and only if n is an odd prime.

If the symmetric representation of sigma(n) has only one polygon (or part), or in other words, if n is a member of A174973 (also of the same sequence A238443) then row n has only a term: T(n,1) = 2 + 2*(A003056(n-1) + A003056(n)). Note that A174973 = A238443 also include all powers of 2 and all even perfect numbers.

			Triangle begins:
   4;
   6;
   4,  4;
  10;
   4,  4;
  12;
   4,  4;
  14;
   4,  6,  4;
   8,  8;
   4,  4;
  18;
   4,  4;
   8,  8;
   4, 12,  4;
  ...
Illustration of row 9:
         4
     _ _ _ _ _
    |_ _ _ _ _|
              |_ _ 6
              |_  |
                |_|_ _
                    | |
                    | |
                    | |  4
                    | |
                    |_|
.
For n = 9 the symmetric representation of sigma(9) has three parts from left to right as follows: a rectangle, a concave hexagon and a rectangle. The number of edges of the polygons are 4, 6, 4 respectively, so the row 9 of the triangle is [4, 6, 4].

Row lengths give A237271.
Row sums give A362818.
Cf. A000079, A000203, A000396, A003056, A065091, A174973, A196020, A235791, A236104, A237270, A237591, A237593, A238443, A244363, A245092, A262626, A274919, A348705.

A362818 Total number of edges of all polygons (or parts) of the symmetric representation of sigma(n).

Original entry on oeis.org

4, 6, 8, 10, 8, 12, 8, 14, 14, 16, 8, 18, 8, 16, 20, 22, 8, 22, 8, 22, 24, 16, 8, 26, 18, 16, 24, 28, 8, 30, 8, 30

Offset: 1

Omar E. Pol, May 04 2023

a(n) = 8 if and only if n is an odd prime.

If the symmetric representation of sigma(n) has only one polygon (or part), or in other words, if n is a member of A174973 (also of the same sequence A238443) then a(n) = 2 + 2*(A003056(n-1) + A003056(n)). Note that A174973 = A238443 also include all powers of 2 and all even perfect numbers.

			Illustration of a(9) = 14:
         4
     _ _ _ _ _
    |_ _ _ _ _|
              |_ _ 6
              |_  |
                |_|_ _
                    | |
                    | |
                    | |  4
                    | |
                    |_|
.
For n = 9 the symmetric representation of sigma(9) has three parts from right to left as follows: a rectangle, a concave hexagon and a rectangle. The number of edges of the polygons are 4, 6, 4 respectively, therefore the total number of edges is 4 + 6 + 4 = 14, so a(9) = 14.

Row sums of A362817.

Cf. A000079, A000203, A000396, A003056, A065091, A174973, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A238443, A244363, A245092, A262626, A274919, A348705.

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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.

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

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A279244 Numbers k with the property that both the smallest and the largest Dyck path of the symmetric representation of sigma(k) share some line segments.

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A348854 a(n) is the total length of all line segments in an octant of the symmetric representation of sigma(n).

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A362817 Irregular triangle read by rows: T(n,k) (n>=1, k>=1) is the number of edges of the k-th polygon (or part), from left to right, of the symmetric representation of sigma(n).

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A362818 Total number of edges of all polygons (or parts) of the symmetric representation of sigma(n).

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