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Statement

The series is the paper's (2.1),

$$\xi=\sum_{n=1}^{\infty}\frac{d(n)}{a_1a_2\cdots a_n},$$

with $d(n)$ the number of divisors of $n$ and the $a_n$ positive integers (p. 638).

Conjecture 2.24 (p. 642). "The series (2.1) is irrational whenever $a_n\to\infty$."

The paper introduces it with "We have not been able to prove the following" (p. 641). The conjecture drops the monotonicity of the section's standing convention: the monotone case is Theorem 2.23, and the condition $a_n\to\infty$ excludes the paper's rational example $a_n=d(n)+1$ (p. 638). The paper declines to pose the analogue for $\varphi(n)$ or $\sigma(n)$, since $a_n=\varphi(n)+1$ or $\sigma(n)+1$ makes those series equal to $1$ (p. 642).

Source. P. Erdős, E. G. Straus, Some number theoretic results, Pacific J. Math. 36 (1971), no. 3, 635--646; Conjecture 2.24 at the top of p. 642. The copy read is identified on the source card.

Read depth. Claims checked: the statement was read on the page image of p. 642. Nothing here is independently reviewed.

What the paper proves toward it

Theorem 2.23 (nondecreasing sequences with $a_1\ge2$) and Lemma 2.14 (any sequence with $|a_n|>c(\log n)^{3/4}$ for all $n$). A sequence that tends to infinity, is not monotone, and falls below every such bound infinitely often is covered by neither.

Bears on

• #258: the problem's question, with $\tau(n)=d(n)$ and positive integers $a_n\to\infty$, is this conjecture as stated. This page records the conjecture as the paper poses it; what later work claims about it is recorded on the problem's claim pages.