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Statement

Question (p. 2, unnumbered). Let $n_1<n_2<\cdots$ be integers with $\limsup_kn_k/k=\infty$. The paper asks whether

$$\sum_{k=1}^\infty\frac{n_k}{2^{n_k}}$$

is irrational. It records three further points.

• It cannot prove irrationality even under the stronger assumption $n_{k+1}-n_k\to\infty$.
• It has no counterexample under the weaker assumption $\limsup_k(n_{k+1}-n_k)=\infty$: no series of this form with rational sum and $\limsup_k(n_{k+1}-n_k)=\infty$ is known to it.
• It guesses that such a series, rational with $\limsup_k(n_{k+1}-n_k)=\infty$, exists.

Other questions on the same page (p. 2). The paper also asks whether for every integer $a$ there is a finite sequence of integers $a<m_1<\cdots<m_k$ with $a/2^a=\sum_{i=1}^km_i/2^{m_i}$. It then recalls the theorem of Erdős and Straus (its reference [2]) that $\sum_kd(k)/M_k$ is irrational, where $d$ counts divisors, $M_k=n_1\cdots n_k$ and $n_1\le n_2\le\cdots$ tend to infinity; it calls it very likely that $n_k\to\infty$ without monotonicity suffices, and calls it frustrating that it cannot prove $\sum_{n\ge2}1/(n!-1)$ irrational.

Source. P. Erdős, Some problems and results on the irrationality of the sum of infinite series, J. Math. Sci. 10 (1975), 1--7, p. 2. The edition read is identified on the source card.

Read depth. Claims checked: the question and the remarks around it were read clause by clause on the printed page. The paper proves nothing about them.

Proof pointer

None: the paper poses the questions and proves nothing about them.

Dependencies

None in the paper.

Bears on

• Problem 260: the problem asks the same question for increasing sequences with $a_n/n\to\infty$. That hypothesis implies $\limsup a_n/n=\infty$, so a yes to this question would answer Problem 260 yes; the paper answers neither. The case $n_{k+1}-n_k\to\infty$ that the paper could not prove is the case of Erdős's 1981 theorem.
• Problem 247: the problem asks about $\sum_n2^{-a_n}$ under the same growth hypothesis $\limsup a_n/n=\infty$, a different series; this question does not address it.
• Problem 68: the paper states on this page that it cannot prove $\sum_{n\ge2}1/(n!-1)$ irrational, the problem's question; it gives no result on it.