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Powers of 2 and even perfect numbers.
4

%I #42 Sep 30 2023 15:15:07

%S 1,2,4,6,8,16,28,32,64,128,256,496,512,1024,2048,4096,8128,8192,16384,

%T 32768,65536,131072,262144,524288,1048576,2097152,4194304,8388608,

%U 16777216,33550336,33554432,67108864,134217728,268435456,536870912,1073741824,2147483648,4294967296,8589869056,8589934592

%N Powers of 2 and even perfect numbers.

%C Numbers k such that the symmetric representation of sigma(k) has only one part, and apart from the central width, the rest of the widths are 1's.

%C Note that the above definition implies that the central width of the symmetric representation of sigma(k) is 1 or 2. For powers of 2 the central width is 1. For even perfect numbers the central width is 2 (see example).

%e Illustration of initial terms:

%e . _ _ _ _ _ _ _ _

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

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

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

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

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

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

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

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

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

%e . _| | | | | |

%e . _| _| | | | |

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

%e . | _ _| | | | |

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

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

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

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

%e . _ _| _ _| | |

%e . | _| _ _| |

%e . _| _| | _ _|

%e . | _| _| |

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

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

%e . | | | _ _|

%e . | | _ _ _| |

%e . | | | _ _ _|

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

%e . 28 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |

%e . | |

%e . | |

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

%e . 32 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|

%e .

%e The diagram shows the first eight terms of the sequence. The symmetric representation of sigma has only one part, and apart from the central width, the rest of the widths are 1's.

%e A317307(n) is the area (or the number of cells) in the n-th region of the diagram.

%Y Union of A000079 and A000396 assuming there are no odd perfect numbers.

%Y Subsequence of A174973.

%Y Cf. A249351 (the widths).

%Y Cf. A317307(n) = sigma(a(n)).

%Y Cf. A000203, A000225, A139256, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A262626.

%K nonn,easy

%O 1,2

%A _Omar E. Pol_, Aug 23 2018