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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.
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%I #45 Dec 09 2021 03:15:57

%S 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 78,80,84,88,90,96,100,104,108,112,120,126,128,132,136,140,144,150,

%U 156,160,162,168,176,180,192,196,198,200,204,208,210,216,220,224,228,234,240,252,256

%N 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 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 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

%e 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 llustration of initial terms:

%e n

%e . _ _ _ _ _ _ _ _ _

%e 1 |_| | | | | | | | | | | | | |

%e 2 |_ _|_| | | | | | | | | | | |

%e 3 |_ _| _|_| | | | | | | | | |

%e 4 |_ _ _| _|_| | | | | | | |

%e _ _ _| _| _ _|_| | | | | |

%e 6 |_ _ _ _| _| | _ _|_| | | |

%e _ _ _ _| |_ _|_| _ _| | |

%e 8 |_ _ _ _ _| _| | _ _ _| |

%e _ _ _ _ _| | _| | _ _ _|

%e 10 |_ _ _ _ _ _| _ _| _| |

%e _ _ _ _ _ _| | _| _|

%e 12 |_ _ _ _ _ _ _| _ _| _|

%e | _ _|

%e | |

%e _ _ _ _ _ _ _ _| |

%e 16 |_ _ _ _ _ _ _ _ _|

%e ...

%Y UNION of A174973 and A262259.

%Y Indices of nonzero terms in A279228.

%Y Complement is A279244.

%Y Cf. A000203, A008586, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A348705.

%K nonn

%O 1,2

%A _Omar E. Pol_, Dec 08 2016