A390085 Half the length of the longest irreducible zero-sum subsets of {-2*n+1, -2*n+3, ..., -3, -1, 1, 3, ..., 2*n-3, 2*n-1}.

1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7

Offset

1,4

Comments

A zero-sum integer set is called irreducible if it does not contain any smaller zero-sum subset.

There may be more than one pair (positive-negative pairs for length greater than 2) of the longest irreducible zero-sum subsets of {-2*n+1, -2*n+3, ..., 2*n-1}, but it remains uncertain whether one of these subsets includes 2*n-1 or -2*n+1.

Links

Table of n, a(n) for n=1..30.

Hu Junhua, Balanced samples of the initial segment of natural numbers Chinese Math R & D BBS, Apr 21 2010.

Formula

a(n)<=sqrt(2n+4)-1.

Example

For n = 4, a(4) = 2, since the longest irreducible zero-sum subsets of {-7, -5, -3, -1, 1, 3, 5, 7} are +-{1, -3, -5, 7}, half its length is 2.
For n = 5, a(5) = 2, since total of {1, 3, 5, 7, 9} is odd.
For n = 6, a(6) = 3, since the longest irreducible zero-sum subsets of {-11, -9, ..., -3, -1, 1, 3, ..., 9, 11} are +-{-1, -3, -5, 7, -9, 11}.
Maximal sets for terms with increased value:
  a(1) = 1: {1, -1}
  a(4) = 2: +-{7, -5, -3, 1}
  a(6) = 3: +-{11, -9, 7, -5, -3, -1}
  a(11) = 4: +-{21, -19, 17, -15, 11, -7, -5, -3} (from Christian Sievers, Nov 12 2025)
             +-{21, -19, 17, -15, -13, 5, 3, 1}
             +-{21, 19, -17, 15, -13, -11, -9, -5}
  a(16) = 5: +-{31, 29, 27, -25, 23, -21, -19, -17, -15, -13}
  a(23) = 6: +-{45, 43, 41, 39, -37, 35, -33, -31, -29, -27, -25, -21}
  a(30) = 7: +-{59, 57, 55, 53, 51, -49, 47, -45, -43, -41, -39, -37, -35, -33}

Crossrefs

Cf. A389802, A389804, A389807, A390083.

Sequence in context: A111393 A323665 A062537 * A279596 A224458 A097688

Adjacent sequences: A390082 A390083 A390084 * A390086 A390087 A390088

Keyword

nonn,hard,more

Author

Hu Junhua, Nov 03 2025

Extensions

a(11) corrected by, and a(15)-a(30) from Christian Sievers, Nov 12 2025

Status

approved