OPEN

Is
\[\sum_n \frac{\phi(n)}{2^n}\]
irrational? Here $\phi$ is the Euler totient function.

The decimal expansion of this sum is A256936 on the OEIS.

OPEN

Let $k\geq 1$ and $\sigma_k(n)=\sum_{d\mid n}d^k$. Is
\[\sum \frac{\sigma_k(n)}{n!}\]
irrational?

OPEN

Let $a_n\to \infty$. Is
\[\sum_{n} \frac{d(n)}{a_1\cdots a_n}\]
irrational, where $d(n)$ is the number of divisors of $n$?

OPEN

Let $a_n$ be a sequence such that $a_n/n\to \infty$. Is the sum
\[\sum_n \frac{a_n}{2^{a_n}}\]
irrational?

This is true under either of the stronger assumptions that

- $a_{n+1}-a_n\to \infty$ or
- $a_n \gg n\sqrt{\log n\log\log n}$.

OPEN

Suppose $a_1<a_2<\cdots$ is a sequence of integers such that for all integer sequences $t_n$ with $t_n\geq 1$ the sum
\[\sum_{n=1}^\infty \frac{1}{t_na_n}\]
is irrational. How slowly can $a_n$ grow?

OPEN

Let $a_n$ be a sequence of integers such that for every sequence of integers $b_n$ with $b_n/a_n\to 1$ the sum
\[\sum\frac{1}{b_n}\]
is irrational. Is $a_n=2^{2^n}$ such a sequence? Must such a sequence satisfy $a_n^{1/n}\to \infty$?

OPEN

Let $a_n$ be a sequence of integers such that for every bounded sequence of integers $b_n$ (with $a_n+b_n\neq 0$ for all $n$ and $b_n$ is not identically zero) the sum
\[\sum \frac{1}{a_n+b_n}\]
is irrational. Are $a_n=2^n$ or $a_n=n!$ examples of such a sequence?

A possible definition of an 'irrationality sequence' (see also [262] and [263]). One example is $a_n=2^{2^n}$. In [ErGr80] they also ask whether such a sequence can have polynomial growth, but Erdős later retracted this in [Er88c], claiming 'It is not hard to show that it cannot increase slower than exponentially'.

Kovač [Ko24c] has proved that $2^n$ is not such an irrationality sequence. More generally, he proves that any strictly increasing sequence of positive integers such that $\sum\frac{1}{a_n}$ converges and for which there exists $C>1$ such that $\frac{1}{a_n^2}\leq C\sum_{k>n}\frac{1}{a_k^2}$ is not such an irrationality sequence.

OPEN

How fast can $a_n\to \infty$ grow if
\[\sum\frac{1}{a_n}\quad\textrm{and}\quad\sum\frac{1}{a_n-1}\]
are both rational?

Cantor observed that $a_n=\binom{n}{2}$ is such a sequence. If we replace $-1$ by a different constant then higher degree polynomials can be used - for example if we consider $\sum_{n\geq 2}\frac{1}{a_n}$ and $\sum_{n\geq 2}\frac{1}{a_n-12}$ then $a_n=n^3+6n^2+5n$ is an example of both series being rational.

Kovač [Ko24c] constructs a sequence $a_n$ with this property which grows exponentially with $n$: \[a_n > 1.01^n.\]

OPEN

Let $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ be the Fibonacci sequence. Let $n_1<n_2<\cdots $ be an infinite sequence with $n_{k+1}/n_k \geq c>1$. Must
\[\sum_k\frac{1}{F_{n_k}}\]
be irrational?

OPEN

Let $P$ be a finite set of primes with $\lvert P\rvert \geq 2$ and let $\{a_1<a_2<\cdots\}=\{ n\in \mathbb{N} : \textrm{if }p\mid n\textrm{ then }p\in P\}$. Is the sum
\[\sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]},\]
where $[a_1,\ldots,a_n]$ is the lowest common multiple of $a_1,\ldots,a_n$, rational or irrational?

If $P$ is infinite this sum is always irrational.

OPEN

Let $f(n)\to \infty$ as $n\to \infty$. Is it true that
\[\sum_n \frac{1}{(n+1)\cdots (n+f(n))}\]
is irrational?

Erdős and Graham write 'the answer is almost surely in the affirmative if $f(n)$ is assumed to be nondecreasing'. Even the case $f(n)=n$ is unknown, although Hansen [Ha75] has shown that
\[\sum_n \frac{1}{\binom{2n}{n}}=\sum_n \frac{n!}{(n+1)\cdots (n+n)}=\frac{1}{3}+\frac{2\pi}{3^{5/2}}\]
is transcendental.