A243980 - OEIS

%I #33 Oct 22 2023 00:36:48

%S 4,16,32,60,84,132,164,224,276,348,396,508,564,660,756,880,952,1108,

%T 1188,1356,1484,1628,1724,1964,2088,2256,2416,2640,2760,3048,3176,

%U 3428,3620,3836,4028,4392,4544,4784,5008,5368,5536,5920,6096,6432,6744,7032,7224,7720

%N Four times the sum of all divisors of all positive integers <= n.

%C Also number of "ON" cells at n-th stage in a structure which looks like a simple 2-dimensional cellular automaton (see example). The structure is formed by the reflection on the four quadrants from the diagram of the symmetry of sigma in the first quadrant after n-th stage, hence the area in each quadrant equals the area of each wedge and equals A024916(n); the sum of all divisors of all positive integers <= n. For more information about the diagram see A237593 and A237270.

%H Indranil Ghosh, <a href="/A243980/b243980.txt">Table of n, a(n) for n = 1..7342</a>

%F a(n) = A016742(n) - 4*A004125(n) = 4*A024916(n).

%F a(n) = 2*(A006218(n) + A222548(n)) = 2*A327329(n). - _Omar E. Pol_, Sep 25 2019

%e Illustration of the structure after 16 stages (contains 880 ON cells):

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%e .

%t Accumulate[4*DivisorSigma[1,Range[50]]] (* _Harvey P. Dale_, May 13 2018 *)

%o (Python)

%o from math import isqrt

%o def A243980(n): return -(s:=isqrt(n))**2*(s+1) + sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))<<1 # _Chai Wah Wu_, Oct 21 2023

%Y Partial sums of A239050.

%Y Partial sums give A244050.

%Y Cf. A000203, A000290, A004125, A016742, A024916, A175254, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934.

%K nonn

%O 1,1

%A _Omar E. Pol_, Jun 18 2014