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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.

%I A279029 #79 Jun 12 2025 22:35:22
%S A279029 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,
%T A279029 78,80,84,88,90,96,100,104,108,112,120,126,128,132,136,140,144,150,
%U A279029 156,160,162,168,176,180,192,196,198,200,204,208,210,216,220,224,228,234,240,252,256
%N 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.
%C A279029 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.
%C A279029 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
%C A279029 From _Hartmut F. W. Hoft_, Jan 25 2025: (Start)
%C A279029 The following three statements are equivalent for numbers k >= 1:
%C A279029 (1) The symmetric representation of sigma(k) is formed of 2 parts that meet at the diagonal.
%C A279029 (2) A249223(k, A003056(k)) = 0 is the only 0 in row k of the triangle, and A237591(k, A003056(k)) = 1.
%C A279029 (3) Row k of the triangle in A341969 contains a single 0 at the center position.
%C A279029 The following two statements are equivalent for numbers k >= 1:
%C A279029 (1) The symmetric representation of sigma(k) consists of a single part.
%C A279029 (2) Row k of the triangle in A249223 contains no 0. (End)
%C A279029 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
%C A279029 My two links below give detailed proofs for the last comment.  - _Hartmut F. W. Hoft_, Jun 10 2025
%H A279029 Antti Karttunen, <a href="/A279029/b279029.txt">Table of n, a(n) for n = 1..20000</a>
%H A279029 Hartmut F. W. Hoft, <a href="/A279029/a279029.pdf">Proof: Dyck Paths of SRS(n) meeting only at diagonal</a>
%H A279029 Hartmut F. W. Hoft, <a href="/A279029/a279029_1.pdf">Proofs for A279029 = A174973 union A262259</a>
%F A279029 a(n) = A323648(n-1) + 1, for n >= 2. - _Hartmut F. W. Hoft_, Jan 25 2025
%e A279029 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.
%e A279029 Illustration of initial terms:
%e A279029   n
%e A279029   .      _ _ _ _   _   _   _   _       _
%e A279029   1     |_| | | | | | | | | | | |     | |
%e A279029   2     |_ _|_| | | | | | | | | |     | |
%e A279029   3     |_ _|  _|_| | | | | | | |     | |
%e A279029   4     |_ _ _|    _|_| | | | | |     | |
%e A279029          _ _ _|  _|  _ _|_| | | |     | |
%e A279029   6     |_ _ _ _|  _| |  _ _|_| |     | |
%e A279029          _ _ _ _| |_ _|_|    _ _|     | |
%e A279029   8     |_ _ _ _ _|  _|     |    _ _ _| |
%e A279029          _ _ _ _ _| |      _|   |  _ _ _|
%e A279029   10    |_ _ _ _ _ _|  _ _|    _| |
%e A279029          _ _ _ _ _ _| |      _|  _|
%e A279029   12    |_ _ _ _ _ _ _|  _ _|  _|
%e A279029                         |  _ _|
%e A279029                         | |
%e A279029          _ _ _ _ _ _ _ _| |
%e A279029   16    |_ _ _ _ _ _ _ _ _|
%e A279029   ...
%t A279029 (* Function path[] is defined in A237270 *)
%t A279029 a279029Q[n_] := Length[Select[Transpose[{Take[path[n], {2,-2}], path[n-1]}], #[[1]]==#[[2]]&]]<=1
%t A279029 a279029[n_] := Select[Range[n], a279029Q]
%t A279029 a279029[256]
%t A279029 (* Alternate, faster function based on A249223 *)
%t A279029 a003056[n_] := Floor[(Sqrt[8n+1]-1)/2]
%t A279029 t249223[n_] :=FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, a003056[n]]]
%t A279029 a262259Q[n_] := Position[t249223[n], 0]=={{a003056[n]}}&&Last[t237591[n]]==1
%t A279029 a174973Q[n_] := !MemberQ[t249223[n], 0]
%t A279029 a279029[n_] := Select[Range[n], a262259Q[#]||a174973Q[#]&]
%t A279029 a279029[256] (* _Hartmut F. W. Hoft_, Jan 25 2025 *)
%o A279029 (PARI) is_A279029 = A379968; \\ _Antti Karttunen_, Jan 12 2025
%Y A279029 UNION of A174973 and A262259.
%Y A279029 Positions of 0's in A279228.
%Y A279029 Complement is A279244.
%Y A279029 Cf. A000203, A003056, A008586, A174973, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A249223, A262259, A323648, A348705, A379968 (characteristic function).
%Y A279029 Cf. A240542, A298856.
%K A279029 nonn
%O A279029 1,2
%A A279029 _Omar E. Pol_, Dec 08 2016