A323648 -id:A323648 - 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

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 *)

A325300 a(n) is the number of faces of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

6, 9, 15, 20, 24, 31, 35, 42, 49, 59, 63, 72, 76, 84, 95, 106, 110, 121, 125

Offset: 1

Omar E. Pol, Apr 16 2019

To calculate a(n) consider that levels greater than n do not exist.
The shape of the n-th level of the pyramid allows us to know if n is prime (see the Formula section).
For more information about the sequences that we can see in the pyramid see A262626.

			For n = 1 the first level of the stepped pyramid (starting from the top) is a cube, and a cube has six faces, so a(1) = 6.

Omar E. Pol, Perspective view of the pyramid (first 16 levels)

Cf. A325301 (number of edges), A325302 (number of vertices).

Cf. A196020, A215848, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A262626, A299692, A323645, A323648.

a(n) = A325301(n) - A325302(n) + 2 (Euler's formula).
a(n) = A323645(n) + 3.
a(n) = a(n-1) + 4 iff n is a prime > 3 (A215848).

A325301 a(n) is the number of edges of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

12, 21, 36, 51, 63, 84, 96, 117, 138, 165, 177, 204, 216, 240, 273, 306, 318, 351, 363

Offset: 1

Omar E. Pol, Apr 16 2019

To calculate a(n) consider that levels greater than n do not exist.

			For n = 1 the first level of the stepped pyramid (starting from the top) is a cube, and a cube has 12 edges, so a(1) = 12.

Omar E. Pol, Perspective view of the pyramid (first 16 levels)

Cf. A325300 (number of faces), A325302 (number of vertices).

Cf. A196020, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A262626, A294848, A294849, A317109, A323648.

a(n) = A325300(n) + A325302(n) - 2 (Euler's formula).

A325302 a(n) is the number of vertices of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

8, 14, 23, 33, 41, 55, 63, 77, 91, 108, 116, 134, 142, 158, 180, 202, 210, 232, 240

Offset: 1

Omar E. Pol, Apr 16 2019

To calculate a(n) consider that levels greater than n do not exist.

			For n = 1 the first level of the stepped pyramid (starting from the top) is a cube, and a cube has 8 vertices, so a(1) = 8.

Omar E. Pol, Perspective view of the pyramid (first 16 levels)

Cf. A325300 (number of faces), A325301 (number of edges).

Cf. A196020, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A262626, A294723, A323648.

a(n) = A325301(n) - A325300(n) + 2 (Euler's formula).

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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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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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A325300 a(n) is the number of faces of the stepped pyramid with n levels described in A245092.

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A325301 a(n) is the number of edges of the stepped pyramid with n levels described in A245092.

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A325302 a(n) is the number of vertices of the stepped pyramid with n levels described in A245092.

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