A391592 - OEIS

%I #27 Jan 18 2026 23:10:13

%S 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,

%T 20,20,21,21,22,23,23,24,24,25,26,27,28,28,29,29,30,31,31,32,33,33,34,

%U 35,36,36,37,37

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

%C This sequence arises from Erdős Problem #321.

%C 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.

%C 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.

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

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

%H Thomas Bloom, <a href="https://www.erdosproblems.com/forum/thread/321">Problem 321</a>, Erdős Problems.

%H Cong Lu, <a href="https://github.com/conglu1997/erdos_321_computation">Python code for generating terms</a>

%H Terence Tao, <a href="https://github.com/teorth/erdosproblems/issues/161">Further computation of R(N) in #321</a>, GitHub Issue #161 (2025).

%e For n=6, the set {1, 2, 3, 4, 5} has all distinct reciprocal sums, so a(6) >= 5.

%e If we attempt to include 6, we find that 1/2 = 1/3 + 1/6.

%e Thus, we must exclude an element from {2, 3, 6}, leading to a maximum size of 5.

%Y Cf. A384927.

%K nonn,more

%O 1,2

%A _Cong Lu_, Jan 10 2026