OFFSET
1,3
COMMENTS
A proper subset of [n]={1..n} is called balanced if its mean value is also (n+1)/2, the overall mean value of [n]. A balanced subset is referred to as a reducible balanced subset if it contains a smaller balanced subset, otherwise, it is called an irreducible balanced subset.
LINKS
Christian Sievers, Table of n, a(n) for n = 1..75
Chinese BBS, Balanced samples of the initial segment of natural numbers (in Chinese).
EXAMPLE
For n=6, a(6)=3, since [6] has 3 irreducible balanced subsets: {1,6}, {2,5}, {3,4}.
For n=7, a(7)=6, since [7] has 6 irreducible balanced subsets: {4}, {1,7}, {2,6}, {3,5}, {1,5,6}, {2,3,7}.
PROG
(Python)
from itertools import combinations
def A389802(n):
if n<3: return 1
b, c = [], 0
for i in (range(1, n) if n&1 else range(2, n, 2)):
j = i*(n+1)
for p in combinations(range(1, n+1), i):
if sum(p)<<1 == j:
q = set(p)
for r in b:
if r <= q:
break
else:
b.append(q)
c += 1
return c # Chai Wah Wu, Oct 23 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Hu Junhua, Oct 15 2025
EXTENSIONS
a(29)-a(36) from Sean A. Irvine, Oct 21 2025
a(37)-a(39) from Chai Wah Wu, Oct 23 2025
a(40)-a(75) from Christian Sievers, Oct 25 2025
STATUS
approved
