23 problems

Density conjecture for sums of distinct powers. If every pair $a_1,a_2\in A$ satisfies $\gcd\{a_1,a_2\}=1$ and

Monotonicity conjecture for $S_3^{(3)}$. The sequence

Let $BS(m,n)$ denote four sequences $A,B,C,D$ of $71$ of lengths $m,m,n,n$ with zero combined non-periodic autocorrelation, and let $BS(n+1,n)$ be the corresponding case. Base sequ…

Let $p>3$ be a non-Mersenne prime, and let $E_n(1,x)$ denote the reversed Dickson polynomial of the second kind. Consider the sequence of values $E_n(1,\frac14)$ modulo $p$. Non-Me…

Let $G$ be an abelian group. For a subset $S\subseteq G\setminus\{0\}$, a sequencing is an ordering whose partial sums are distinct, while a rotational sequencing permits the initi…

Let $n_1,n_2,n_3,\ldots$ be the sequence defined by $\sigma_{n_\ell}=\ell+1$ for all $\ell\geq 1$, where the values of $\sigma_n$ arise from the paper's preceding construction. Quo…

The conjectures. (A) For every integer $k\ge0$, the set $\{n\ge1:a_n=k\}$ has a positive density $\delta_k$, and

Logarithmic-phase asymptotic conjecture. For all $n>0$,

Sign-change conjecture. The sequence $(c_n)_{n\geq0}$ changes sign infinitely often.

Growth conjecture for $AP_k$-covering sequences. (i) For every integer $k\geq 3$, there exists an $AP_k$-covering sequence $A$ such that

Non-periodicity conjecture. Rows of Table 1 are not ultimately periodic.

Bounded-row conjecture. Given $a\geq 1$, the nim-sequence $(\mathcal{G}(0,a,n))_{n\geq a}$ is bounded.

Let $a_n|_{n=d4a}$ be a sequence evaluated at infinity, and let $z$ denote a relation in $*G$. Infinite-sequence monotonicity claim. The monotonicity of $a_n|_{n=d4a}$ can be deter…

Starting from the normalized initial values $0<x<1$, let $(x_4,x_5,\ldots)$ be the associated M&m sequence, and let $L(x)$ denote the minimum integer $k$ such that $x_n=x_k$ for al…

Let $(t_n)_{n=1}^{\infty}$ and $(T_n)_{n=1}^{\infty}$ be the sequences whose values are listed in the table preceding the conjecture. Monotonicity conjecture. For every…

Let $\{d_{\ell}(m)\}$ be the sequence described above, and define the operator … A sequence is infinitely logconcave if $\mathfrak{L}^j(\{x_k\})$ is nonnegative for every…

Sequence existence conjecture. There exists an infinite sequence on four letters, $A,B,C$ and $D$ such that the sequence is nonrepetitive, palindrome-free and avoids the subsequenc…

Let $b\geq 2$ be an integer, and let $\mathfrak{S}_b(n)$ denote the last non-zero digit of $n^n$ in base $b$. Periodicity conjecture. The sequence $\mathfrak{S}_b(n)$ is eventually…

Euler-transform coefficient conjecture. The sequence $(c_m)_{m=1}^{\infty}$ is strictly positive, increasing, strictly convex, and satisfies $\Delta^k c\geq0$ for $k=3,4$; explicit…

Let $\left\{a_n^{(\omega)}\right\}_{n=1}^{\infty}$ range over all possible sequences, and let $p\tilde{\lim}a_n^{(\omega)}$ denote the set of approximated partial limits while the…

Yang's conjecture. $NN(n)\ne\emptyset$ for all positive even integers $n$.

Let $\mathbb{Z}_2$ be the $2$-adic integers, let $\widetilde{Q}_i:\mathbb{Z}_2\to\mathbb{Z}_2$ be the continuous extensions of $Q_i(n)=Q(3n+i)$, and let $c_k$ be the unique zero of…

For an integer $x$ and prime $p$, let $[x]_p$ denote the largest power of $p$ dividing $x$, with $[0]_p=\infty$. Let $A(n)$ be the factorial-numerator sequence from the Taylor part…