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Statement

Problem E2689 (Hahn, reprinted on printed p. 224 at the head of the solution; proposed in Amer. Math. Monthly 85 (1978), p. 47). Is there a nonempty finite set $S$ of positive integers such that

1. every $n\in S$ has $n-1\in S$ or $n+1\in S$, and
2. $\sum_{n\in S}1/n$ is an integer?

Condition 1 means that $S$ is a union of maximal runs of consecutive integers, each of length at least two; maximal runs are separated by at least one integer not in $S$, since two adjacent runs would be one run. The problem fixes neither the number of runs nor the integer in condition 2.

Solution (Montgomery, p. 224). Yes. The sets

$$S_1=\{1,2,7,8,13,14,39,40,76,77,285,286\}$$

and

$$S_2=\{2,3,4,5,6,7,9,10,17,18,34,35,84,85\}$$

both satisfy the two conditions, and in each case the reciprocals sum to $2$. As runs, $S_1$ is the six blocks $[1,2]$, $[7,8]$, $[13,14]$, $[39,40]$, $[76,77]$, $[285,286]$, each of length two, and $S_2$ is the five blocks $[2,7]$, $[9,10]$, $[17,18]$, $[34,35]$, $[84,85]$. The page adds, quoted: "The second example was also found by Dean Hickerson."

In the problem's notation. Writing $\sum_{n\in I}1/n$ for a block $I$, $S_2$ gives $2=\sum_{i=1}^5\sum_{n\in I_i}1/n$ with $I_1=[2,7]$, $I_2=[9,10]$, $I_3=[17,18]$, $I_4=[34,35]$, $I_5=[84,85]$, five intervals that are distinct, pairwise non-overlapping and pairwise non-adjacent, each of length at least two: the integer $2$ has a representation of the shape Problem 289 asks for with $k=5$. $S_1$ gives one with $k=6$ and every block of length exactly two. Neither says anything about the integer $1$ or about all large $k$.

Source. L.-S. Hahn (proposer) and Peter L. Montgomery (solver), E2689, Amer. Math. Monthly 86 (1979), no. 3, p. 224, doi:10.2307/2321534; printed p. 224 = PDF p. 2 of the JSTOR scan, read on the page image (the text layer garbles the displayed condition 2). The copy read is identified in the source digest.

Read depth. Claims checked: the reprinted problem, both sets, the closing sentence and the note on Hickerson were read clause by clause on the page image on 2026-09-22. The page prints no derivation of the sets. Both reciprocal sums were recomputed here in exact rational arithmetic and equal $2$; the check is the page's only content beyond the statement, and nothing here is independently reviewed.

Proof pointer

The page states the two sets and that their Egyptian fractions sum to $2$, with no argument. The claim is a finite computation: with exact rational arithmetic, $\sum_{n\in S_1}1/n=2$ and $\sum_{n\in S_2}1/n=2$, and conditions 1 and 2 are read off the listed elements.

Dependencies

None; the statement is a finite verification.

Bears on

• Problem 289: the primary source of the five-interval representation of $2$ that the site's commentary and the 1980 monograph (p. 34) cite; it fixes the attribution (Montgomery's solution, with the second set also found by Hickerson) and shows that Hahn's problem asked for any integer under the separated-block condition with the number of blocks free. The problem's status is unchanged.