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%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
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