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Source. Theorem 5.1, preprint p. 8; proof pp. 8--9; Remark pp. 9--10; Theorem 5.2 and Example 5.1, pp. 10--11. Read on the rendered pages.

Statement

Theorem 5.1 (p. 8): "Let $\{a_n\}_{n=1}^{\infty}$ be a monotonic sequence of positive integers satisfying $p_n=o(a_n^2)$. Then $S=\sum_{n=1}^{\infty}\frac{p_n}{a_1\ldots a_n}$ is rational if and only if $\frac{p_n}{a_n-1}$ is constant for $n\ge n_0$."

The paper notes (p. 2) that $p_n=o(a_n^2)$ forces $a_n>\sqrt{n\log n}$ for large $n$, and (pp. 2 and 8) that this drops the condition $\liminf a_n/p_n=0$ of Erdős–Straus 1974, Theorem 3.1, replacing it by the necessary condition that $p_n/(a_n-1)$ is not eventually constant. Compared with the 1958 theorem of Erdős (nondecreasing $q_n$ with $q_n(\log n)^k/n\to\infty$), the growth hypothesis is much weaker and the rational values are the same family, since $p_n/(a_n-1)=1/q$ constant means $a_n=qp_n+1$.

Proof structure (pp. 8--9)

Suppose $S=r/q$; then $qS_n\in\mathbb{Z}$ for all $n$ (Lemma 2.1), and (3) gives $|S_n-p_n/a_n|<\epsilon$ for $n\ge n_1(\epsilon)$. If $S_n<S_{n+1}$ then $p_{n+1}/a_{n+1}-p_n/a_n>1/q-2\epsilon$, so $p_{n+1}-p_n>(1/q-2\epsilon)a_n$, which by $p_n\ll n\log n$ and $a_n/\sqrt n\to\infty$ happens for $O(\sqrt N\log N)$ indices in $[N,2N)$. If $S_{n-1}=S_n<S_{n+1}$ then $p_{n-1}\mid(a_{n-1}-1)$, so $p_{n-1}<a_{n-1}\le a_n$, and with $p_{n+1}-p_n=o(p_{n-1})$ this cannot occur for large $n$. If $S_n>S_{n+1}$ for more than $N/2$ indices in $[N,2N)$, then (5) $a_{n+1}-a_n>(1/q-2\epsilon)a_na_{n+1}/p_n$ makes $a_{2N}\gg N^2/\log N$, and Theorem 3.2 gives the theorem in that case. Otherwise $S_n=S_{n+1}$ for arbitrarily large $n$; if $p_n/(a_n-1)$ is not eventually constant, then $S_n$ is not eventually constant, and the pattern $S_{n-1}=S_n>S_{n+1}>\cdots>S_{n+k}<S_{n+k+1}$ with $0<k\le q$ occurs infinitely often; it forces $a_{n+k}\ge(1/(4q)-2\epsilon)(n+k)\log(n+k)$ and then $p_{n+k+1}-p_{n+k}>p_{n+k}/(20q^2)$, contradicting $p_{n+1}-p_n=o(p_n)$. $\blacksquare$

The prime inputs are $p_n\ll n\log n$, the lower bound $p_{n-1}>\frac12n\log n$ in the last step (p. 9), $p_{n+1}-p_n=o(p_n)$ and the primality used in "$p_{n-1}\mid(a_{n-1}-1)$".

Remark on monotonicity (pp. 9--10)

With $b_n=p_n$ and $a_n=p_n+1$ the partial sums are $1-1/(a_1\ldots a_N)$, so the sum is $1$; replacing infinitely often a pair $(a_N,a_{N+1})$ by $(p_N+2,(p_{N+1}+1)/2)$ keeps the limit $1$ and the growth order of $a_n$ but destroys monotonicity. So the monotonicity hypothesis cannot be dropped from the theorem; the paper does not treat the non-monotone expectation of Erdős 1988 (p. 103), for which a 2026 note claims a counterexample with $a_n=o(p_n)$.

Further results in the section

Theorem 5.2 (p. 10) replaces the primality of $p_n$ by a gcd condition on monotonic positive integers $b_n$ with $b_n=o(a_n^2)$ and $a_{2n}b_{2n}=o(na_n^2)$; Example 5.1 (p. 11): $\sum_{n\ge1}(p_n/n!)^k$ is irrational for every integer $k\ge1$ (a different series from $\sum p_n^k/n!$).

Bears on. #251 (the monotone relatives of the problem's series; the theorem excludes bounded $a_n$ through $p_n=o(a_n^2)$).