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

1,2

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.

Links

Table of n, a(n) for n=1..63.

Formula

a(n) = A279029(n+1) - 1, for n >= 1. - Hartmut F. W. Hoft, Jan 25 2025

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 *)

Crossrefs

Cf. A174973, A000203, A196020, A236104, A237270, A237271, A237591, A237593, A244145, A245092, A245192, A262259, A262626, A279029, A279244.

Sequence in context: A139282 A375985 A228121 * A353037 A238526 A281541

Adjacent sequences: A323645 A323646 A323647 * A323649 A323650 A323651

Keyword

nonn

Author

Omar E. Pol, Apr 02 2019

Extensions

a(17)-a(63) by Hartmut F. W. Hoft, Jan 25 2025

Status

approved