A378718 a(n) is the smallest integer k such that [1,2,...,k] contains a set of n but no more disjoint consecutive subsequences, each having element sum k.

1, 3, 15, 27, 75, 165, 345, 315, 693, 945, 1365, 2457, 2835, 4455, 3465, 4095, 7245, 15435, 12285, 10395, 17325, 27405, 24255, 29925, 31185, 48195, 70875, 45045, 58905, 61425, 89775, 128205, 142065, 121275, 135135

Offset

1,2

Comments

We observe that a(n) == 0 (mod 3) for n>1.

Links

Table of n, a(n) for n=1..35.

Example

  +----+-----+------------------------------------------------+
  | n  |   k |     n subsets whose sum of elements = k        |
  +----+-----+------------------------------------------------+
  | 1  |   1 | {1}                                            |
  +----+-----+------------------------------------------------+
  | 2  |   3 | {1,2}, {3}                                     |
  +----+-----+------------------------------------------------+
  | 3  |  15 | {1,2,3,4,5}, {7,8}, {15}                       |
  +----+-----+------------------------------------------------+
  | 4  |  27 | {2,3,4,5,6,7}, {8,9,10}, {13,14}, {27}         |
  +----+-----+------------------------------------------------+
  | 5  |  75 | {3,4,5,6,7,8,9,10,11,12}, {13,14,15,16,17},    |
  |    |     | {24,25,26}, {37,38}, {75}                      |
  +----+-----+------------------------------------------------+
  | 6  | 165 | {4,5,...,18}, {25,26,...,30}, {31,32,33,34,35},|
  |    |     |   {54,55,56}, {82,83}, {165}                   |
  +----+-----+------------------------------------------------+
  | 7  | 345 | {4,5,...,26}, {30,31,...,39}, {55,56,...60},   |
  |    |     | {67,68,...71}, {114,115,116}, {172,173},{345}  |
  +----+-----+------------------------------------------------+
Note: There are several solutions to reach n disjoint subsets, for example for n=3, we can choose from one of the 2 solutions: {{4,5,6},{7,8},{15}} or {{1,2,3,4,5}, {7,8}, {15}}.

Crossrefs

Sequence in context: A347536 A002259 A050848 * A077785 A222291 A015646

Adjacent sequences: A378715 A378716 A378717 * A378719 A378720 A378721

Keyword

nonn,more

Author

Michel Lagneau, Dec 05 2024

Extensions

a(10)-a(35) from Alois P. Heinz, Dec 06 2024

Status

approved