2 problems

Apéry-sequence consecutive negativity conjecture. For every integer $N\geq2$, there is a positive integer $n$ such that

Apéry-sequence Brun criterion conjecture. There is an unbounded subsequence of positive integers $n_k\to\infty$ such that $\delta_{n_k}<0$. The sequence $y_n/x_n$ is increasing and…