A391592 Maximum size of a subset S of {1..n} such that all subset sums of {1/k : k in S} are distinct.

1, 2, 3, 4, 5, 5, 6, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 15, 15, 16, 17, 17, 18, 19, 20, 20, 21, 21, 22, 23, 23, 24, 24, 25, 26, 27, 28, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 35, 36, 36, 37, 37

Offset

1,2

Comments

This sequence arises from Erdős Problem #321.

It is the maximum size of a subset S of {1..n} such that the sums of the reciprocals of the elements of any subset of S are all distinct.

Equivalently, there are no two disjoint subsets X, Y of S such that Sum_{x in X} 1/x = Sum_{y in Y} 1/y.

Terms 1..20 computed by epistemologist; terms 21..36 computed by Stijn Cambie; terms 37..54 computed by Cong Lu.

The sequence first diverges from A384927 at n = 21 (see Tao link).

Links

Table of n, a(n) for n=1..54.

Thomas Bloom, Problem 321, Erdős Problems.

Cong Lu, Python code for generating terms

Terence Tao, Further computation of R(N) in #321, GitHub Issue #161 (2025).

Example

For n=6, the set {1, 2, 3, 4, 5} has all distinct reciprocal sums, so a(6) >= 5.
If we attempt to include 6, we find that 1/2 = 1/3 + 1/6.
Thus, we must exclude an element from {2, 3, 6}, leading to a maximum size of 5.

Crossrefs

Cf. A384927.

Sequence in context: A392332 A006163 A331268 * A384927 A390394 A053757

Adjacent sequences: A391589 A391590 A391591 * A391593 A391594 A391595

Keyword

nonn,more

Author

Cong Lu, Jan 10 2026

Status

approved