A288772 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.

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

Offset

1,2

Comments

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

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

Links

Table of n, a(n) for n=1..81.

Example

Figures A..D show the evolution of the table of partitions into consecutive parts described in A286000, for n = 8..11:
.     ---------------------------------------------------------------------
Figure:      A            B                    C                  D
.     ---------------------------------------------------------------------
.    n:      8            9                   10                 11
Row   ---------------------------------------------------------------------
1     |  1;        |  1;             |   1;             |   1;            |
1     |  2;        |  2;             |   2;             |   2;            |
3     |  3,  2;    |  3,  2;         |   3,  2;         |   3,  2;        |
4     |  4,  1;    |  4,  1;         |   4,  1;         |   4,  1;        |
5     |  5,  3;    |  5,  3;         |   5,  3;         |   5,  3;        |
6     |  6,  2,  3;|  6,  2,  3;     |   6,  2,  3;     |   6,  2,  3;    |
7     |  7,  4,  2;|  7,  4,  2;     |   7,  4,  2;     |   7,  4,  2;    |
8     | [8], 3,  1;|  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];|  13,  7,  4,  1;|
.     ---------------------------------------------------------------------
. a(n):      8              11                13                 13
.     ---------------------------------------------------------------------
For n = 8 we need a table with at least 8 rows, so a(8) = 8.
For n = 9 we need a table with at least 11 rows, so a(9) = 11.
For n = 10 we need a table with at least 13 rows, so a(10) = 13.
For n = 11 we need a table with at least 13 rows, so a(11) = 13.

Crossrefs

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

Sequence in context: A239826 A246601 A288529 * A053196 A159634 A186690

Adjacent sequences: A288769 A288770 A288771 * A288773 A288774 A288775

Keyword

nonn

Author

Omar E. Pol, Jun 17 2017

Status

approved