6 problems

Sixfold strict increase conjectures. The sequences $r_6(n)$ and $s_6(n)$ satisfy

For a sequence $\mathcal{S}$, let a step be an integer $N$ at which its maximum order complexity changes from $M(\mathcal{S},N-1)$ to $M(\mathcal{S},N)$, and define the ratio of su…

Let $\mathcal{T}_P$ denote the Thue–Morse sequence along a polynomial subsequence, where $P$ is a polynomial of degree $d$. Write $M(\mathcal{T}_P,N)$ for its maximum order complex…

Let $\mathbf{t}=t_0t_1t_2\cdots$ be the Thue–Morse sequence, defined by $t_0=0$, $t_{2n}=t_n$, and $t_{2n+1}=1-t_n$ for $n\geq 0$. Let $f(n)$ denote the number of unbordered factor…

Let $b,r$ be positive integers with $\gcd(b-1,r)=1$, and let $c$ be any integer. Write $s_b(n)$ for the sum of the base-$b$ digits of $n$. Bounded-offset conjecture. There exists a…

Periodicity conjecture. The sequence $\{\gamma_a(n)\}$ is periodic. If $2^m\mathrel{\|}a$, then its minimal period has $2^{m+1}$ terms. If $a$ is evil, this period consists of the…