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.
1, 2, 2, 4, 4, 5, 5, 8, 8, 8, 9, 9, 11
Offset: 1
Examples
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: . ------------------------------------------------------ Figure: A B C ------------------------------------------------------------ . n = 11 12 13 Row ------------------------------------------------------ 1 | 1; | 1; | 1; | 1 | 2; | 2; | 2; | 3 | 3, 2; | 3, 2; | 3, 2; | 4 | 4, 1; | 4, 1; | 4, 1; | 5 | 5, 3; | 5, 3; | 5, 3; | 6 | 6, 2, 3; | 6, 2, 3; | 6, 2, 3; | 7 | 7, 4, 2; | 7, 4, 2; | 7, 4, 2; | 8 | 8, 3, 1; | 8, 3, 1; | 8, 3, 1; | 9 | [9],[5],[4]; | [9],[5],[4]; | 9, 5, 4; | 10 | 10, [4],[3], 4;| 10, [4],[3], 4;| 10, 4, 3; 4;| 11 | 11, 6, [2], 3;| 11, 6, [2]; 3;| [11],[6], 2, 3;| 12 | | 12, 5, 5, 2;| 12, [5], 5, 2;| 13 | | | 13, 7, 4, 1;| . ------------------------------------------------------ . a(n): 9 9 11 . ------------------------------------------------------ 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. 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. 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.
Comments