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A288773 a(n) is the largest of all positive integers whose partitions into consecutive parts can be totally represented in the first n rows of the table described in A286000.

%I A288773 #23 Oct 21 2017 21:06:42
%S A288773 1,2,2,4,4,5,5,8,8,8,9,9,11
%N A288773 a(n) is the largest of all positive integers whose partitions into consecutive parts can be totally represented in the first n rows of the table described in A286000.
%C A288773 a(n) has the same definition related to the table A286001 which is another version of the table A286000.
%C A288773 First differs from A288774 at a(12), which shares infinitely many terms.
%e A288773 Figures A, B, C show the evolution of the table of partitions into consecutive parts described in A286000, with 11, 12 and 13 rows respectively:
%e A288773 .     ------------------------------------------------------
%e A288773 Figure:       A                B                  C
%e A288773 ------------------------------------------------------------
%e A288773 .   n =      11               12                 13
%e A288773 Row   ------------------------------------------------------
%e A288773 1     |  1;            |   1;            |   1;            |
%e A288773 1     |  2;            |   2;            |   2;            |
%e A288773 3     |  3,  2;        |   3,  2;        |   3,  2;        |
%e A288773 4     |  4,  1;        |   4,  1;        |   4,  1;        |
%e A288773 5     |  5,  3;        |   5,  3;        |   5,  3;        |
%e A288773 6     |  6,  2,  3;    |   6,  2,  3;    |   6,  2,  3;    |
%e A288773 7     |  7,  4,  2;    |   7,  4,  2;    |   7,  4,  2;    |
%e A288773 8     |  8,  3,  1;    |   8,  3,  1;    |   8,  3,  1;    |
%e A288773 9     | [9],[5],[4];   |  [9],[5],[4];   |   9,  5,  4;    |
%e A288773 10    | 10, [4],[3], 4;|  10, [4],[3], 4;|  10,  4,  3;  4;|
%e A288773 11    | 11,  6, [2], 3;|  11,  6, [2]; 3;| [11],[6], 2,  3;|
%e A288773 12    |                |  12,  5,  5,  2;|  12, [5], 5,  2;|
%e A288773 13    |                |                 |  13,  7,  4,  1;|
%e A288773 .     ------------------------------------------------------
%e A288773 . a(n):       9                 9                 11
%e A288773 .     ------------------------------------------------------
%e A288773 For n = 11, in the first 11 rows of the table can be represented the partitions into consecutive parts of the integers 1, 2, 3, 4, 5, 6, 7, 8 and 9. The largest of these positive integers is 9, so a(11) = 9.
%e A288773 For n = 12, in the first 12 rows of the table can be represented the partitions into consecutive parts of the integers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 11. The largest of these positive integers is 11, but the partitions into consecutive parts of 10 cannot be represented, so a(12) = 9, not 11.
%e A288773 For n = 13, in the first 13 rows of the table can be represented the partitions into consecutive parts of the integers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11. The largest of these positive integers is 11, so a(13) = 11.
%Y A288773 Cf. A237593, A286000, A286001, A288529, A288772, A288774.
%K A288773 nonn,more
%O A288773 1,2
%A A288773 _Omar E. Pol_, Jun 17 2017