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A245092 The even numbers (A005843) and the values of sigma function (A000203) interleaved. 153

%I

%S 0,1,2,3,4,4,6,7,8,6,10,12,12,8,14,15,16,13,18,18,20,12,22,28,24,14,

%T 26,24,28,24,30,31,32,18,34,39,36,20,38,42,40,32,42,36,44,24,46,60,48,

%U 31,50,42,52,40,54,56,56,30,58,72,60,32,62,63,64,48

%N The even numbers (A005843) and the values of sigma function (A000203) interleaved.

%C Consider an irregular step pyramid with n steps. The base of the pyramid is equal to the symmetric representation of A024916(n), the sum of all divisors of all positive integers <= n. Two of the faces of the pyramid are the same as the representation of the n-th triangular numbers as a staircase. The total area of the pyramid is equal to 2*A024916(n) + A046092(n). The volume is equal to A175254(n). By definition a(2n-1) is A000203(n), the sum of divisors of n. Starting from the top a(2n-1) is also the total area of the horizontal part of the n-th step of the pyramid. By definition, a(2n) = A005843(n) = 2n. Starting from the top, a(2n) is also the total area of the irregular vertical part of the n-th step of the pyramid.

%C On the other hand the sequence also has a symmetric representation in two dimensions, see Example.

%C From _Omar E. Pol_, Dec 31 2016: (Start)

%C We can find the pyramid after the following sequences: A196020 --> A236104 --> A235791 --> A237591 --> A237593.

%C The structure of this infinite pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593 (see the links).

%C The terraces at the m-th level of the pyramid are also the parts of the symmetric representation of sigma(m), m >= 1, hence the sum of the areas of the terraces at the m-th level equals A000203(m).

%C Note that the step pyramid is also one of the 3D-quadrants of the step pyramid described in A244050.

%C For more information about the pyramid see A237593 and all its related sequences. (End)

%H Robert Price, <a href="/A245092/b245092.txt">Table of n, a(n) for n = 0..20000</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr01.jpg">A pyramid related to the divisor function and other integers sequences</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr02.jpg">Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the pyramid (first 16 levels)</a>

%F a(2*n-1) + a(2n) = A224880(n).

%e Illustration of initial terms:

%e ----------------------------------------------------------------------

%e a(n) Diagram

%e ----------------------------------------------------------------------

%e 0 _

%e 1 |_|\ _

%e 2 \ _| |\ _

%e 3 |_ _| | |\ _

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

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

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

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

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

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

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

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

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

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

%e 14 \ _ _ _ _| | _| | _ _| | | | | | | | | |\ _

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

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

%e 13 |_ _ _ _ _| | _| _| _ _ _| | | | | | | | | | |

%e 18 \ _ _ _ _ _| | _| _| _ _| | | | | | | | | |

%e 18 |_ _ _ _ _ _| | _| | _ _|_| | | | | | | |

%e 20 \ _ _ _ _ _ _| | _| | _ _ _| | | | | | |

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

%e 22 \ _ _ _ _ _ _| | _ _| _|_| _ _ _|_| | | |

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

%e 24 \ _ _ _ _ _ _ _| | _| | _| | _ _ _| |

%e 14 |_ _ _ _ _ _ _| | | _| _| _| | _ _ _|

%e 26 \ _ _ _ _ _ _ _| | |_ _| _| _| |

%e 24 |_ _ _ _ _ _ _ _| | _ _| _| _|

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

%e 24 |_ _ _ _ _ _ _ _| | | _ _|

%e 30 \ _ _ _ _ _ _ _ _| | |

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

%e 32 \ _ _ _ _ _ _ _ _ _|

%e ...

%e a(n) is the total area of the n-th set of symmetric regions in the diagram.

%e .

%e From _Omar E. Pol_, Aug 21 2015: (Start)

%e The above structure contains a hidden pattern, simpler, as shown below:

%e Level _ _

%e 1 _| | |_

%e 2 _| _|_ |_

%e 3 _| | | | |_

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

%e 5 _| | _|_ | |_

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

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

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

%e 9 _| | | _|_ | | |_

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

%e 11 _| | _| | | | |_ | |_

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

%e 13 _| | | _| | |_ | | |_

%e 14 _| _| _| | _|_ | |_ |_ |_

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

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

%e ...

%e The symmetric pattern emerges from the front view of the step pyramid.

%e Note that starting from this diagram A000203 is obtained as follows:

%e In the pyramid the area of the k-th vertical region in the n-th level on the front view is equal to A237593(n,k), and the sum of all areas of the vertical regions in the n-th level on the front view is equal to 2n.

%e The area of the k-th horizontal region in the n-th level is equal to A237270(n,k), and the sum of all areas of the horizontal regions in the n-th level is equal to sigma(n) = A000203(n).

%e (End)

%e From _Omar E. Pol_, Dec 31 2016: (Star)

%e Illustration of the top view of the pyramid with 16 levels:

%e .

%e n A000203 A237270 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

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

%e 2 3 = 3 |_ _|_| | | | | | | | | | | | | |

%e 3 4 = 2 + 2 |_ _| _|_| | | | | | | | | | | |

%e 4 7 = 7 |_ _ _| _|_| | | | | | | | | |

%e 5 6 = 3 + 3 |_ _ _| _| _ _|_| | | | | | | |

%e 6 12 = 12 |_ _ _ _| _| | _ _|_| | | | | |

%e 7 8 = 4 + 4 |_ _ _ _| |_ _|_| _ _|_| | | |

%e 8 15 = 15 |_ _ _ _ _| _| | _ _ _|_| |

%e 9 13 = 5 + 3 + 5 |_ _ _ _ _| | _|_| | _ _ _|

%e 10 18 = 9 + 9 |_ _ _ _ _ _| _ _| _| |

%e 11 12 = 6 + 6 |_ _ _ _ _ _| | _| _| _|

%e 12 28 = 28 |_ _ _ _ _ _ _| |_ _| _|

%e 13 14 = 7 + 7 |_ _ _ _ _ _ _| | _ _|

%e 14 24 = 12 + 12 |_ _ _ _ _ _ _ _| |

%e 15 24 = 8 + 8 + 8 |_ _ _ _ _ _ _ _| |

%e 16 31 = 31 |_ _ _ _ _ _ _ _ _|

%e ... (End)

%t Table[If[EvenQ@ n, n, DivisorSigma[1, (n + 1)/2]], {n, 0, 65}] (* or *)

%t Transpose@ {Range[0, #, 2], DivisorSigma[1, #] & /@ Range[#/2 + 1]} &@ 65 // Flatten (* _Michael De Vlieger_, Dec 31 2016 *)

%Y Cf. A000203, A004125, A024916, A005843, A175254, A196020, A224880, A235791, A236104, A237270, A237271, A237591, A237593, A239050, A239660, A239931-A239934, A243980, A244050, A244360-A244363, A244370, A244371, A244970, A244971, A245093, A261350, A262626, A277437, A279387, A280223, A280295.

%K nonn

%O 0,3

%A _Omar E. Pol_, Jul 15 2014

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Last modified November 19 06:03 EST 2019. Contains 329310 sequences. (Running on oeis4.)