Characterizing the sequences of rationals realized by partitions of the unit fractions into finite subsets (Erdős–Stein, 1963)

Erdős and Stein (1963) proved that the set of unit fractions {1, 1/2, 1/3, ...} can be partitioned into disjoint finite subsets whose sums are precisely the positive rationals, and likewise into blocks whose sums are precisely the positive integers. They asked for a necessary and sufficient condition on a sequence of positive rationals to arise as the block sums of such a partition. The question remains open: apart from the two existence families and scattered special cases, no characterization (candidate condition, or proof that a natural condition is necessary or sufficient) is known.

Background

A unit fraction (Egyptian fraction) is a fraction of the form $1/n$ with $n$ a positive integer; a classical theorem going back to Fibonacci and Sylvester states that every positive rational is a finite sum of distinct unit fractions. The research area asks finer structural questions about such representations: which sets of denominators suffice to represent all rationals (reciprocal bases, introduced by Herbert S. Wilf), how many terms are needed, and how the representations can be organized globally. Erdős and Stein (1963) addressed two questions of Wilf about reciprocal bases for the integers (a set $S$ of positive integers such that every positive integer is a finite sum of reciprocals of distinct elements of $S$): they constructed a reciprocal basis of density zero (that is, $|S \cap \{1, \dots, n\}| = o(n)$ as $n \to \infty$) for the rationals (strengthened in their Theorem 4: for every sequence $(s_i)$ with divergent reciprocal sum there is a reciprocal basis $(n_i)$ with $n_i > s_i$), and they determined that the minimal number $f(n)$ of distinct unit fractions needed to represent the integer $n$ satisfies $\lim f(n)e^{-n} = e^{-\gamma}$ with $\gamma$ Euler's constant (independently proved by P. J. van Albada and J. H. van Lint). Their Theorem 2 is the departure point of the present problem: the whole set of unit fractions $\{1, 1/2, 1/3, \dots\}$ can be partitioned into disjoint finite subsets $S_1, S_2, \dots$ such that each positive rational is the sum of the elements of exactly one $S_i$; the same holds with 'each positive rational' replaced by 'each positive integer'. In other words, one can group the terms of the harmonic series (the divergent series $\sum_{n \geq 1} 1/n$) once and for all into finite blocks whose sums enumerate, exactly and without repetition, all positive rationals — or all positive integers. The proof rests on a lemma (essentially Sylvester's greedy expansion) that every positive rational $r$ admits a finite Egyptian representation whose denominators are all at least any prescribed bound $A$ and are pairwise at least $A$ apart, which lets successive blocks avoid all previously used denominators. Immediately after Theorem 2 the authors remark: 'It would be interesting to know the necessary and sufficient condition that a sequence of rationals $r_1, r_2, r_3, \dots$ corresponds to the sums of a partition of the set of unit fractions into disjoint finite subsets.' The question is genuinely subtle: the divergence of $\sum_i r_i$ is necessary (the harmonic series diverges and every unit fraction must be used) but not sufficient — for instance, each single unit fraction $1/n$ must lie in some block, so the fraction $1 = 1/1$ forces some $r_i \geq 1$, and the constant sequence $r_i \equiv 1/2$, whose sum diverges, is therefore not realizable; the exact interaction between the decreasing sequence of available unit fractions and the prescribed block sums is an exact infinite bin-packing-type constraint (every unit fraction $1/n$ must be packed into exactly one finite block with a prescribed total) with no known general characterization. The problem sits inside the modern cluster of Erdős unit-fraction questions — whether every finite coloring of the positive integers admits one color class containing distinct denominators whose reciprocals sum to 1 (solved by Ernest Croot), whether every set $A$ of natural numbers of positive upper density, i.e. $\limsup_{n \to \infty} |A \cap \{1, \dots, n\}|/n > 0$, contains such a finite set of denominators (solved by Thomas F. Bloom, sharpened by Yang P. Liu and Mehtaab Sawhney), and which denominator sets support enough representations (David Eppstein's denominator sets closed under doubling; the Bleicher–Erdős problem on lacunary, i.e. geometrically growing, denominator sequences, settled by van Doorn and Kovač) — but is distinct from all of these: they concern representations constrained by a denominator set, whereas the Erdős–Stein question concerns exhausting the fixed set of all unit fractions by blocks with prescribed sums.

Problem Statement

Determine a necessary and sufficient condition for a sequence of positive rationals $r_1, r_2, r_3, \dots$ to correspond to the sums of a partition of the set of unit fractions $\{1, 1/2, 1/3, \dots\}$ into disjoint finite subsets; that is, characterize exactly those sequences $(r_i)$ for which there exists a partition of the positive integers into countably many disjoint finite blocks $B_1, B_2, B_3, \dots$ (so that every positive integer $n$ — i.e. every unit fraction $1/n$ — belongs to exactly one block) with

$$\sum_{n \in B_i} \frac{1}{n} = r_i \quad \text{for every } i,$$

where the order of the $r_i$ is immaterial: the partition realizes the sequence as the multiset of its block sums (this is the reading under which Erdős and Stein state their two existence results, which enumerate the positive rationals and the positive integers without a prescribed order). Erdős and Stein (1963) proved existence for the sequence enumerating all positive rationals and for the sequence enumerating all positive integers; the answer for general sequences — including which conditions beyond $r_i > 0$ and $\sum_i r_i = \infty$ (both necessary) are required, and whether any natural extension of these conditions is sufficient — is not known.

References

1. Paul Erdős and Sherman Stein, "Sums of distinct unit fractions", Proc. Amer. Math. Soc. 14 (1963), 126–131, DOI 10.1090/S0002-9939-1963-0142502-9, https://doi.org/10.1090/S0002-9939-1963-0142502-9 (primary source; the open question is the remark immediately following Theorem 2, pp. 127–128; zbMATH Zbl 0115.26501)
2. P. Erdős and E. Szemerédi, "On a problem of P. Erdős and S. Stein", Acta Arith. 15 (1968), no. 1, 85–90, DOI 10.4064/aa-15-1-85-90, https://doi.org/10.4064/aa-15-1-85-90 (solves a different Erdős–Stein problem on disjoint covering systems of arithmetic progressions)
3. R. L. Graham, "A theorem on partitions", J. Austral. Math. Soc. 3 (1963), 435–441, DOI 10.1017/s1446788700039045, https://doi.org/10.1017/s1446788700039045 (every integer n ≥ 78 is a sum of distinct positive integers whose reciprocals sum to 1)
4. R. L. Graham, "On Finite Sums of Unit Fractions", Proc. London Math. Soc. (3) 14 (1964), 193–207, DOI 10.1112/plms/s3-14.2.193, https://doi.org/10.1112/plms/s3-14.2.193 (reciprocal bases from divergent reciprocal sequences)
5. Donald Silberger, "On sums of Egyptian fractions", arXiv:2412.02049 (2024), https://arxiv.org/abs/2412.02049 (partitions the tail [kd, ∞) into finite sets each with reciprocal sum n/d — the constant-sequence special case on a tail)
6. Wouter van Doorn and Vjekoslav Kovač, "Lacunary sequences whose reciprocal sums represent all rational numbers in an interval", Acta Arith. 223 (2026), no. 3, 275–295, DOI 10.4064/aa251001-13-1, arXiv:2509.24971, https://arxiv.org/abs/2509.24971 (disproves the Bleicher–Erdős lacunary conjecture, Erdős Problem #355, https://www.erdosproblems.com/355)
7. Wouter van Doorn, "Partitions with prescribed sum of reciprocals: asymptotic bounds", arXiv:2502.02200 (2025), https://arxiv.org/abs/2502.02200 (near-optimal bounds for Graham's n_{α,m})
8. Donald Silberger, Sylvia Silberger and David Hobby, "Sums of finitely many distinct rationals", arXiv:1702.01316 (2017; v2 2019), https://arxiv.org/abs/1702.01316 (Theorem 1.1: every positive rational admits an infinite pairwise disjoint family of finite Egyptian representations; the predecessor of Silberger 2024)
9. Paul Erdős and Ronald L. Graham, "Old and new problems and results in combinatorial number theory", Monographie No. 28 de L'Enseignement Mathématique, Université de Genève, 1980, 128 pp., zbMATH Zbl 0434.10001, https://zbmath.org/?q=an%3A0434.10001
10. Herbert S. Wilf, "Reciprocal bases for integers" (Research Problem 6), in "Research problems", Bull. Amer. Math. Soc. 67 (1961), 456–461, DOI 10.1090/S0002-9904-1961-10632-0, https://doi.org/10.1090/S0002-9904-1961-10632-0 (the problem announcement that introduced reciprocal bases; Wilf's item is on pp. 456–457)
11. P. J. van Albada and J. H. van Lint, "Reciprocal Bases for the Integers", Amer. Math. Monthly 70 (1963), 170–174, DOI 10.1080/00029890.1963.11990060, https://doi.org/10.1080/00029890.1963.11990060 (reciprocal-bases study; the f(n) asymptotic of Erdős–Stein's Theorem 5)
12. Ernest Croot, "On a coloring conjecture about unit fractions", Ann. of Math. (2) 157 (2003), no. 2, 545–556, DOI 10.4007/annals.2003.157.545, https://doi.org/10.4007/annals.2003.157.545 (solves the Erdős–Graham conjecture that for every finite coloring of the integers, some one color class contains distinct integers whose reciprocals sum to 1)
13. Thomas F. Bloom, "On a density conjecture about unit fractions" (with an appendix by Thomas F. Bloom and Bhavik Mehta), J. Eur. Math. Soc. 27 (2024), no. 11, 4563–4589, DOI 10.4171/jems/1456, arXiv:2112.03726, https://doi.org/10.4171/jems/1456 (every set of natural numbers of positive upper density contains a finite subset whose reciprocals sum to 1, answering a question of Erdős and Graham)
14. Yang P. Liu and Mehtaab Sawhney, "On Further Questions Regarding Unit Fractions", Int. Math. Res. Not. IMRN (2026), no. 2, DOI 10.1093/imrn/rnaf382, arXiv:2404.07113, https://doi.org/10.1093/imrn/rnaf382 (sharpens Bloom's density theorem: a subset A of [1, N] with reciprocal sum at least (log N)^{4/5+o(1)} contains a subset whose reciprocals sum to 1)
15. David Eppstein, "Egyptian fractions with denominators from sequences closed under doubling", J. Integer Seq. 24 (2021), Article 21.8.8, arXiv:2109.12217, https://cs.uwaterloo.ca/journals/JIS/VOL24/Eppstein/eppstein2.html (representations from denominator sets closed under doubling, including the practical numbers)
16. Michael N. Bleicher and Paul Erdős, "Denominators of Egyptian fractions", J. Number Theory 8 (1976), 157–168, DOI 10.1016/0022-314x(76)90098-6, https://doi.org/10.1016/0022-314x(76)90098-6 (with its sequel, Illinois J. Math. 20 (1976), 598–613, DOI 10.1215/ijm/1256049650, the origin of the Bleicher–Erdős lacunary conjecture, Erdős Problem #355, disproved by van Doorn–Kovač)
17. J. J. Sylvester, "On a Point in the Theory of Vulgar Fractions", Amer. J. Math. 3 (1880), 332–335 and 388–389, DOI 10.2307/2369261, https://doi.org/10.2307/2369261 (the greedy algorithm for Egyptian fractions)