4 problems

Prime-increment conjecture. For every $n\geq 2$, we have $b_n\in\{1\}\cup\mathbb{P}$, where $\mathbb{P}$ denotes the set of prime numbers.

Let $a_n$ and $b_n$ be defined by … and $b_n=a_n/a_{n-1}-1$ for $n\geq2$. Cloitre's prime-occurrence conjecture. The sequence $(b_n)$ contains every prime number other than $3$, wh…

Define … and, for $n\geq2$, define $b_n=a_n/a_{n-1}-1$. The sequence $(b_n)$ consists of positive integers. Cloitre's conjecture. For any $n\geq2$, $b_n$ is either $1$ or a prime n…

Let $a(n)$ be the sequence defined by the recurrence in the paper, and let $n_1\geq 1$ satisfy $a(n_1)\geq 1$. Transience conjecture. There exists an $N$ such that $a(n)-a(n-1)$ is…