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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).

%I A323648 #36 Feb 07 2025 19:38:23
%S A323648 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,
%T A323648 79,83,87,89,95,99,103,107,111,119,125,127,131,135,139,143,149,155,
%U A323648 159,161,167,175,179,191,195,197,199,203,207,209,215,219,223,227,233,239,251,255
%N 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).
%C A323648 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.
%C A323648 a(2) = 2 is the only even number in the sequence.
%C A323648 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.
%F A323648 a(n) = A279029(n+1) - 1, for n >= 1. - _Hartmut F. W. Hoft_, Jan 25 2025
%t A323648 (* Function path[] is defined in A237270 *)
%t A323648 a323648Q[n_] := Length[Select[Transpose[{Take[path[n+1], {2,-2}], path[n]}], #[[1]]==#[[2]]&]]<=1
%t A323648 a323648[n_] := Select[Range[n], a323648Q]
%t A323648 a323648[255]
%t A323648 (* Functions a262259Q[ ] and a174973Q[ ] are defined in A279029 *)
%t A323648 a323648[n_] := Select[Range[n], a262259Q[#+1]||a174973Q[#+1]&]
%t A323648 a323648[255] (* _Hartmut F. W. Hoft_, Jan 25 2025 *)
%Y A323648 Cf. A174973, A000203, A196020, A236104, A237270, A237271, A237591, A237593, A244145, A245092, A245192, A262259, A262626, A279029, A279244.
%K A323648 nonn
%O A323648 1,2
%A A323648 _Omar E. Pol_, Apr 02 2019
%E A323648 a(17)-a(63) by _Hartmut F. W. Hoft_, Jan 25 2025