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).

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

Author

Omar E. Pol, Apr 02 2019

Keywords

• nonn

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