A390084 Length of the longest irreducible balanced subsets of {-n..n}.

1, 2, 2, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16

Offset

0,2

Comments

A zero-sum subset of {-n..n} is called irreducible if it does not contain any smaller zero-sum subset.

Links

Table of n, a(n) for n=0..50.

Chinese BBS, Balanced samples of the initial segment of natural numbers (in Chinese).

Example

For n = 3, a(3) = 3, since the longest irreducible balanced subsets of {-3..3} are +-{1,2,-3}.
For n = 4, a(4) = 4, since the longest irreducible balanced subsets of {-4..4} are +-{-1, 2, 3, -4}.
From Bert Dobbelaere, Nov 09 2025: (Start)
Maximal sets for terms with increased value:
  a(0) = 1: {0}
  a(1) = 2: {-1, 1}
  a(3) = 3: {3, -2, -1}
  a(4) = 4: {4, -3, -2, 1}
  a(6) = 5: {6, -5, 4, -3, -2}
  a(8) = 6: {8, -7, 6, -5, -4, 2}
  a(11) = 7: {11, 10, -9, -8, -7, 2, 1}
  a(14) = 8: {14, 13, -12, -11, -10, 3, 2, 1}
  a(18) = 9: {18, 17, 16, -15, -14, -13, -12, 2, 1}
  a(21) = 10: {21, 20, 19, -18, -17, -16, -15, 3, 2, 1}
  a(25) = 11: {25, 24, 23, -22, -21, -20, -19, 4, 3, 2, 1}
  a(30) = 12: {30, 29, 28, 27, -26, -25, -24, -23, -22, 3, 2, 1}
  a(34) = 13: {34, 33, 32, 31, -30, -29, -28, -27, -26, 4, 3, 2, 1}
  a(39) = 14: {39, 38, 37, 36, -35, -34, -33, -32, -31, 5, 4, 3, 2, 1}
  a(45) = 15: {45, 44, 43, 42, 41, -40, -39, -38, -37, -36, -35, 4, 3, 2, 1}
  a(50) = 16: {50, 49, 48, 47, 46, -45, -44, -43, -42, -41, -40, 5, 4, 3, 2, 1}
(End)

Crossrefs

Cf. A389802, A389804, A389807, A390082.

Sequence in context: A227737 A194177 A348386 * A385072 A023190 A047783

Adjacent sequences: A390081 A390082 A390083 * A390085 A390086 A390087

Keyword

nonn,more

Author

Hu Junhua, Oct 28 2025

Extensions

a(15)-a(50) from Bert Dobbelaere, Nov 09 2025

Status

approved