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a(n) is the minimum number of rows from the table described in A286000 that are required to represent the partitions of all positive integers <= n into consecutive parts.
8

%I #27 Oct 21 2017 21:06:32

%S 1,2,4,4,6,8,8,8,11,13,13,14,14,17,19,19,19,21,21,24,26,26,26,26,29,

%T 29,32,34,34,34,34,34,38,38,41,43,43,43,44,44,44,48,48,51,53,53,53,53,

%U 55,55,56,59,59,62,64,64,64,64,64,67,67,67,71,71,74,76,76,76,76,76,76,80,80,80,84,84,87,89,89,89,89

%N a(n) is the minimum number of rows from the table described in A286000 that are required to represent the partitions of all positive integers <= n into consecutive parts.

%C a(n) has the same definition related to the table A286001 which is another version of the table A286000.

%C First differs from A288529 at a(11), which shares infinitely many terms.

%e Figures A..D show the evolution of the table of partitions into consecutive parts described in A286000, for n = 8..11:

%e . ---------------------------------------------------------------------

%e Figure: A B C D

%e . ---------------------------------------------------------------------

%e . n: 8 9 10 11

%e Row ---------------------------------------------------------------------

%e 1 | 1; | 1; | 1; | 1; |

%e 1 | 2; | 2; | 2; | 2; |

%e 3 | 3, 2; | 3, 2; | 3, 2; | 3, 2; |

%e 4 | 4, 1; | 4, 1; | 4, 1; | 4, 1; |

%e 5 | 5, 3; | 5, 3; | 5, 3; | 5, 3; |

%e 6 | 6, 2, 3;| 6, 2, 3; | 6, 2, 3; | 6, 2, 3; |

%e 7 | 7, 4, 2;| 7, 4, 2; | 7, 4, 2; | 7, 4, 2; |

%e 8 | [8], 3, 1;| 8, 3, 1; | 8, 3, 1; | 8, 3, 1; |

%e 9 | | [9],[5],[4]; | 9, 5, 4; | 9, 5, 4; |

%e 10 | | 10, [4],[3], 4;| [10], 4, 3, [4];| 10, 4, 3; 4;|

%e 11 | | 11, 6, [2], 3;| 11, 6, 2; [3];| [11],[6], 2, 3;|

%e 12 | | | 12, 5, 5, [2];| 12, [5], 5, 2;|

%e 13 | | | 13, 7, 4, [1];| 13, 7, 4, 1;|

%e . ---------------------------------------------------------------------

%e . a(n): 8 11 13 13

%e . ---------------------------------------------------------------------

%e For n = 8 we need a table with at least 8 rows, so a(8) = 8.

%e For n = 9 we need a table with at least 11 rows, so a(9) = 11.

%e For n = 10 we need a table with at least 13 rows, so a(10) = 13.

%e For n = 11 we need a table with at least 13 rows, so a(11) = 13.

%Y Cf. A001227, A109814, A204217, A237593, A286000, A286001, A288529, A288773, A288774.

%K nonn

%O 1,2

%A _Omar E. Pol_, Jun 17 2017