Arithmetic properties of integers in chains and reflections of g-ary expansions

Recently, there has been a sharp rise of interest in properties of digits primes. Here we study yet another question of this kind. Namely, we fix an integer base g ⩾ 2 and then for every infinite sequence

D={d_i}^∞_i=0 ∈ {0,..., g - 1}^∞

of g-ary digits we consider the counting function ω_D,g(N) of integers n ⩽ N for which ∑^{n − 1} _{i = 0}d_igⁱ is prime. We construct sequences D for which ω_D,g(N) grows fast enough, and show that for some constant ϑ_g < g there are at most O(ϑ^N _g) initial elements (d₀, …, d_{N − 1}) of D for which ω_D,g(N) = N + O(1). We also discuss joint arithmetic properties of integers and mirror reflections of their g-ary expansions.