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

Author

Omar E. Pol, Dec 08 2016

Keywords

• nonn

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

• 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

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