Open-problem statements, with their sources attached.

Let $a_1 < a_2 < \dots$ be a sequence of integers such that $\lim_{n\to\infty} \frac{a_n}{a_{n-1}^2} = 1$ and $\sum \frac{1}{a_n} \in \mathbb{Q}$.

Then, for all sufficiently large $n \ge 1$, $a_n = a_{n-1}^2 - a_{n-1} + 1$.

The Flint Hills series summing $csc(n)^2 / n^3$ from $n=1$ to $\infty$ converges. (Note that we 0-index the series below.)

The Cookson Hills series summing $sec(n)^2 / n^3$ from $n=1$ to $\infty$ converges.

Conjecture: Voronovskaja-type formula for Bézier-Bernstein operators with shape parameter $\alpha > 0$, $\alpha \neq 1$.

The source asks for sufficiently smooth functions. This concrete version uses ContDiffOn ℝ 2 f I as a readable baseline regularity assumption; since the domain is the compact interval $[0,1]$, this also explains why no separate boundedness assumption is included here. The variants below record the unknown smoothness threshold more explicitly.

Variant of the Bézier-Bernstein Voronovskaja problem which treats "sufficiently smooth" as an eventual condition in the smoothness order $m$: for all sufficiently large finite $m$, every $C^m$ function on $[0,1]$ should have the asserted asymptotic formula.

Existence-only version of the eventual-smoothness variant. This separates the first part of the source problem, proving that the scaled sequence has some limit, from the stronger task of finding an explicit expression for that limit.

Variant of the Bézier-Bernstein Voronovskaja problem with the required smoothness order itself left as an answer. Replacing (answer(sorry) : ℕ × ((ℝ → ℝ) → ℝ → ℝ)) by a concrete value lets one state the conjecture for a chosen regularity threshold.

Do infinitely many pairs $(a, a+2)$ occur in Ulam's sequence?

Does Ulam's sequence eventually have periodic differences? That is, is $a(n+1) - a(n)$ eventually periodic?

Part (iii), is the density of the sequence 0?