22 problems

A Beatty sequence is a sequence of the form $\{\lfloor n\alpha\rfloor:n\in\mathbb{N}\}$ with modulus $\alpha>1$. A partition of $\mathbb{N}$ into $m$ Beatty sequences has distinct…

Dekking's conjecture. For every integer $n\geq 1$,

Duchêne–Rigo conjecture. There exists an invariant game having

Let $q,m\geq 3$. Let $p_1,p_2,\ldots,p_m$ be distinct integers satisfying $0<p_k<q$, $\gcd(q,p_1,\ldots,p_m)=1$, and such that no proper subset $I\subsetneq\{1,\ldots,m\}$ satisfie…

Quantitative Erdős–Kac conjecture for Beatty sequences. The distance between $X_N$ and the standard Gaussian $\mathcal{N}(0,1)$ is bounded above by

Let $z=1$, let $v=1/\phi=\phi-1$, and define $s_n$ by … where $s_1=1.91041843973790\mathbf{4}$ is the numerically computed value associated with the continued fraction, and let the…

For an integer $j\geq 1$, let $\alpha_j$ be the $j$-th metallic mean, so that $\alpha_j^2=j\alpha_j+1$. Let $r_1$ be the unique positive real number such that all terms defined by…

Kimberling's conjecture. The sequence $(S(n))_{n\geq 1}$ consists precisely of the positive integers not belonging to the Beatty sequence

Frantzikinakis's conjecture. For any such multiplicative functions, one has

Pell-Lucas discrepancy conjecture. The natural numbers $6,12,20,26,34,\dots$ for which

Generalized Beatty conjecture. The positions at which the $1$'s occur in the column of digit $i$ are given by unions of generalized Beatty sequences.

A Beatty sequence is a sequence of the form $\left(\left\lfloor n\alpha+\beta\right\rfloor\right)_{n\geq 1}$ for fixed real parameters $\alpha$ and $\beta$, with the relevant hypot…

Let $\theta>1$ be irrational, let $\psi\in[0,\infty)$, and define the Beatty sequence … For a positive integer $n$, let $P(n)$ denote its largest prime factor, and let $\Psi(X,y)$…

Let $A(n)=\lfloor n\sqrt{2}\rfloor$ and let $AA$ denote the iterated Beatty sequence obtained by applying $A$ to the terms of $A$. A morphic representation of $AA$ is given on an a…

Fraenkel's conjecture. If the system has no multiplicity, then $\alpha_i=p/q_i$, where

Let $a>1$ and $b>1$ be irrational numbers. The Beatty-sequence logarithm bound by $1$. … Numerical data in the source suggest this stronger bound than the preceding conjecture. The…

Let $a>1$ and $b>1$ be irrational numbers. The Beatty-sequence logarithm bound. … This conjecture is motivated by estimating the corresponding series through the quantities…

Let $(A_n,B_n)_{n\ge 1}$ be a pair of non-homogeneous complementary Beatty sequences with $A_1=1$. The sequence $(B_n)_{>0}$ is $B_1$-superadditive when, for all $m,n>0$,…

Beatty-periodicity conjecture. The set $\mathcal{P}(G)$ is periodic if and only if the modulus $\alpha$ of $(a_n)$ is rational. This proposes a complete rational-versus-irrational…

Let $S(\alpha,\beta)$ denote the inhomogeneous Beatty sequence with modulus $\alpha$ and shift $\beta$. Let an exact covering family (EEC) be a family of such sequences whose membe…

Let $d\ge 1$ and let $\bar{\alpha},\bar{\beta}\in[0,1)^d$. For a real vector, write $\langle x_1,\ldots,x_k\rangle$ for the condition that $x_1,\ldots,x_k$ are irrational. Assume t…

Let $\alpha>0$ be an irrational number and let $\beta$ be a real number. The expression $\floor{\alpha p+\beta}$ denotes the largest integer not exceeding $\alpha p+\beta$. Primes…