3 problems

Let $\mathcal{W}_2$ be the set of all positive integers $a$ for which there is no integer $m>1$ such that $(a-2^k,m)>1$ for every positive integer $k$. The $\mathcal{W}_2$ density…

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper, and let $n,r,s\geqslant 1$. Even-exponent Catalan-triangle congruence conjecture. … This is a complementary…

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper, let $r\geqslant 0$ and $s\geqslant 1$, and assume $n\geqslant 4^s-1$. Power-of-four threshold conjecture. ……