%I #46 Oct 16 2023 22:25:21

%S 1,3,7,12,15,28,31,39,42,60,56,72,63,91,90,96,124,120,120,168,127,144,

%T 195,186,224,180,234,252,217,210,280,248,360,312,255,336,336,403,372,

%U 392,378,363,480,372,546,508,399,468,465,504,434,576,600,504,504,560,546,744,728,511

%N Sum of divisors of the n-th number whose divisors increase by a factor of 2 or less.

%C Also consider the n-th number k with the property that the symmetric representation of sigma(k) has only one part. a(n) is the area of the diagram (see the example). For more information see A237593 and its related sequences.

%H Paolo Xausa, <a href="/A317305/b317305.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F a(n) = A000203(A174973(n)).

%e Illustration of initial terms (n = 1..13):

%e .

%e a(n)

%e _ _ _ _ _ _ _ _ _ _ _ _ _

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

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

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

%e 7 |_ _ _| _|_| | | | | | | | | | | | | | | | | |

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

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

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

%e 15 |_ _ _ _ _| _| | _ _ _| | | | | | | | | | | | | |

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

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

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

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

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

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

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

%e 31 |_ _ _ _ _ _ _ _ _| | _ _| _| _ _| | | _ _ _ _ _|_| |

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

%e 39 |_ _ _ _ _ _ _ _ _ _| | _ _| _| _ _| _ _| | |

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

%e 42 |_ _ _ _ _ _ _ _ _ _ _| | _ _ _| _| _| | _ _|

%e | | | _| _| |

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

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

%e 60 |_ _ _ _ _ _ _ _ _ _ _ _ _| | | | _ _|

%e | | _ _ _| |

%e | | | _ _ _|

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

%e 56 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |

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

%e 72 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |

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

%e 63 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|

%e .

%e The length of the largest Dyck path of the n-th diagram equals A047836(n).

%e The semilength equals A174973(n).

%e a(n) is the area of the n-th diagram.

%t A317305[upto_]:=Table[If[AllTrue[Map[Last[#]/First[#]&,Partition[Divisors[n],2,1]],#<=2&],DivisorSigma[1,n],Nothing],{n,upto}];

%t A317305[500] (* _Paolo Xausa_, Jan 12 2023 *)

%Y A317307 is a subsequence.

%Y Cf. A174973.

%Y Cf. A000203, A047836, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A262626, A361208.

%K nonn

%O 1,2

%A _Omar E. Pol_, Aug 25 2018