Conjectures on the recursively defined prime-counting sequence

Let $(s_n)_{n\ge1}$ be the sequence defined in the paper, let $a_n=s_n-s_{n-1}$ for $n>1$, and for an integer $b\ge2$ define

$$A(b)=\sum_{n\ge1}a_nb^{-n}.$$

The conjectures. (A) For every integer $k\ge0$, the set $\{n\ge1:a_n=k\}$ has a positive density $\delta_k$, and

$$\delta_1>\delta_0>\delta_2>\delta_3>\delta_4>\cdots.$$

(B) One has

$$\liminf_{n\to\infty}(n-s_n)<\infty.$$

(C) For every integer $b\ge2$, the real number $A(b)$ is transcendental. These conjectures describe the anticipated distribution and arithmetic nature of the increments of the recursively defined sequence; the preceding results establish only the limiting relations $a_n/n\to0$ and $s_n/n\to1$, leaving (A)--(C) open.