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For every integer $n\ge1$,

$$R_n\ge2^{\lceil n/2\rceil}\ge2^{n/2}.$$

More precisely, every subset of $U_n=\{\lfloor n/2\rfloor+1,\ldots,n\}$ has reciprocal sum at most one. This is a lower bound for the relaxed count $R_n$, not for the exact count $E_n$.

Proof. Put $m=\lfloor n/2\rfloor$. The set $U_n$ has $n-m=\lceil n/2\rceil$ elements, and $n-m\le m+1$. Every denominator in $U_n$ is at least $m+1$, so

$$\sum_{i\in U_n}\frac1i \le\frac{n-m}{m+1}\le1.$$

Every subset has no larger reciprocal sum. The $2^{n-m}$ distinct subsets of $U_n$ are therefore all counted by $R_n$. This includes $n=1$, where $U_1=\{1\}$. $\square$

Source and endpoint refinement. Steinerberger, arXiv:2403.17041v5, p. 1, paragraph after the Theorem. The source writes the upper half informally as $\{n/2,n/2+1,\ldots,n\}$. That notation has a nonintegral endpoint when $n$ is odd and includes a problematic extra term for some small even $n$: for $n=4$, the reciprocal sum over $\{2,3,4\}$ is $13/12$. The family $U_n$ above supplies an explicit version valid for every $n\ge1$. This refinement does not affect the source's eventual upper bound.

Read depth. Claims checked: the remark and its bound $2^{n/2}$ were read on p. 1.

Bears on. #297 only as a limit on the relaxation: an upper bound for $E_n$ proved by bounding $R_n$ is at least $2^{\lceil n/2\rceil}$. No lower bound for $E_n$ follows.