15 problems

Arnold's sequence conjecture. Arnold's sequence $(u_k)_{k\geq1}$ is the $f$-transform of the quadruple $(2,4,4,4)$.

Let $b_m(n)$ be the sequence in the paper and let $m$ be a fixed positive integer. For each $n\in\mathbb{N}$ and sufficiently large $k$ (written $k\gg1$), suppose there is a non-de…

Let $b_m(n)$ be the sequence considered in the paper and let $m\in\mathbb{N}_{\geq2}$ be such that $m$ is not of the form $2^k-1$ for any $k\in\mathbb{N}_{+}$. Unbounded valuation…

Let $\nu_2$ denote the 2-adic valuation, and let $t_m(n)$ be the coefficient sequence in the paper. For $n\in\mathbb{N}$, … and … More generally, for each $k\in\mathbb{N}_2$ there…

The odd-argument 2-adic valuation conjecture. If $0\le k<2^{e-1}$, then

The 2-adic binomial reciprocal-sum valuation conjecture. If $0\le k<2^e$, then

Power-of-two valuation formula conjecture. If $t\geq3$, then

Power-of-two valuation conjecture. If $n=2^t$, then

The third approximation claim. For most $m\in\mathbb{N}$,

The second approximation conjecture.

Define $\lambda_p(m)=\frac12\left(1-(-1)^{m\bmod p}\right)$ and … Define … and $I_1=\{x_1(m):m\geq0\}$. The first approximation conjecture. … unless $m\in I_1$. The formula is an e…

For each level $m$, let $C_{m,j}=\{q_{2^m i+j}:i\in\mathbb{N}\}$ denote the surviving non-constant class in the branch beginning with indices congruent to $0$ modulo $4$, and let…

For $n\in\mathbb{N}$, let $q_n=\nu_{2}(S(n,5))$, and call $i$ exceptional when $q_{4i}\neq q_{4i+3}$. The observed exceptional indices begin with $7,39,71,103,\ldots$. The exceptio…

Main conjecture. The first index for which there is a constant class is $m_{0}-1$: every class is part of the $j$-level for $1\leq j\leq m_{0}-2$. For every $m\geq m_{0}$, the $m$-…

Initial-segment symmetry conjecture. These graphs have a symmetry property generated by an initial segment: the rest of each graph is determined by adding a central piece followed…