A386820 a(n) is the size of largest subset of {1, 1/2, ..., 1/n} that can be partitioned into two parts, the sum of elements of which are equal.

0, 0, 0, 0, 0, 4, 4, 4, 4, 4, 4, 6, 6, 6, 9, 9, 9, 11, 11, 11, 14, 14, 14

Offset: 1

Author

Yifan Xie, Aug 03 2025

Keywords

• nonn
• more

Examples

			For a(6) = 4, the set {1, 1/2, 1/3, 1/6} is chosen because 1 = 1/2 + 1/3 + 1/6.
The two parts for a(18) = 11 are 1 + 1/5 + 1/6 + 1/15 = 1/2 + 1/3 + 1/4 + 1/9 + 1/10 + 1/12 + 1/18.
The two parts for a(20) = 11 are 1 + 1/9 + 1/10 + 1/15 + 1/18 = 1/2 + 1/3 + 1/5 + 1/6 + 1/12 + 1/20.
		

Links

• Art of Problem Solving, 2020 GQMO Problem 4, which shows that a(n) >= 2*n/5 for sufficiently large n.
• Vincent Jugé, Proof that a(n) > c*n for all real numbers c < 1 and sufficiently large n

Crossrefs

Cf. A001008, A098181.

Formula

a(n) = a(n-1) if n/k = p is a prime and p > A001008(k).