24 problems

Let $S(A)=(a_n)$ be an independent Stanley sequence: there are constants $\lambda=\lambda(A)$ and $\kappa=\kappa(A)$ such that for all $k\ge\kappa$ and $0\le i<2^k$,…

Let $S(A)=(a_n)$ be a Stanley sequence, meaning the sequence generated greedily from a finite $3$-free set $A\subset\mathbb N_0$. Rolnick's growth-rate conjecture. For all $n$ larg…

Let $S(A)=(a_n)$ be a Stanley sequence, meaning the greedy sequence generated from a finite 3-free set $A$ of nonnegative integers. Let $\alpha$ denote the constant appearing in th…

Completion conjecture. Every 3-free set has a completion.

Let $A_4$ be the Stanley sequence starting from the set $\{0,4\}$. Its growth rate is compared with the scale $n^2/\log n$. The conjecture for $A_4$. The sequence does not have gro…

Let $S_i$ be the greedily constructed 3-free sequence indexed by $i$. Construct the infinite grid whose zeroth row consists of strings containing only digits $0$ and $1$, and whose…

Let the Stanley cross-sequence be the sequence formed by the first terms of the greedily constructed 3-free sequences $S_i$, and represent non-negative integers using digits $0$,…

A set $A$ is near-modular modulo $3^{\ell+1}$ if it has the near-modularity property used in the paper. Near-modular set conjecture. For every $\ell\in\mathbb{Z}^{+}$, there exists…

Eventual $p$-cardinality conjecture. For every odd prime $p$, there exists a natural number $n_p$ such that for every $n\ge n_p$, there is a modular $p$-free set $A$ with

$p$-density conjecture. There exists a modular $p$-Stanley sequence $S_p(A)=(a_n)$ such that

Non-power cardinality conjecture. For every $n\in\mathbb{N}\cup\{0\}$, there exists a modular $p$-free set $A$ such that

Eventual cardinality conjecture. There exists $n_0\in\mathbb{N}$ such that for every $n\ge n_0$, there are an integer $N_n\in\mathbb{N}$ and a modular set $A_n$ modulo $N_n$ satisf…

Density conjecture. There exists a modular sequence $S(A)=(a_n)$ such that

Odlyzko–Stanley conjecture. For $p=3$, the only $m$ for which $S_3(0,m)$ is well-behaved, with modularity as the relevant notion of well-behavedness, are

Pseudomodularity conjecture. A Stanley sequence follows Type 1 growth if and only if it is pseudomodular.

Rolnick's regularity conjecture. Every Stanley sequence following Type 1 growth is regular.

Let $\lambda$ be the parameter associated with an independent Stanley sequence $S(A)$. Rolnick's conjecture. For every integer $\lambda$ other than $1,3,5,9,11,15$, there exists an…

A Stanley sequence $S(A)=\{a_n\}$ is generated by the greedy construction from a finite 3-free set $A$ of nonnegative integers. The two-type growth conjecture. Every Stanley sequen…

Let $S(A)$ be a dependent Stanley sequence, meaning a regular Stanley sequence that is not independent. An element of $S(A)$ is deletable if deleting it produces another dependent…

Generalized shift conjecture. Then $S_k(c,A)$ is defined and is a regular Stanley sequence with core $S(A)$. This would strengthen the preceding construction theorem for shifted St…

For a regular Stanley sequence $S(A)$, let $\lambda(A)$ denote its character. The range of the character function is the set of values attained by $\lambda(A)$ over regular Stanley…

Let $S(A)=\{a_n\}$ and $S(A')=\{a'_n\}$ be Stanley sequences. Say that $S(A)$ is faithful to $S(A')$ if, for every $a'_n<\omega(A')$, there is an index $m$ such that $a_m=a'_n$. Fo…

Type 2 growth conjecture. Every irregular Stanley sequence follows Type 2 growth. The paper proves that regular Stanley sequences follow Type 1 growth and identifies this assertion…

Let $S(A)=\{a_n\}$ be a regular Stanley sequence. For sufficiently large $k$, write … where $\sigma(A)$ is the shift index and $\beta(A)$ is the associated rational parameter. Nonn…