9 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 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…

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 conjecture. If $n=2^t$, then

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

The second approximation conjecture.

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 $m$-…