The prime-power formula for the minimal period of the generalized digit map

Let $M = \prod_{p|M} p^{a_p}$ and $N = K \cdot \prod_{p|M} p^{b_p}$ be integers with $a_p, b_p \ge 1$ and $\gcd(K,M) = 1$. For each prime factor $p|M$, define

$$t_p:= \lceil a_p / b_p \rceil.$$

The sequence $b(n)$ is assumed to have a minimal period $P_{\min}$. Minimal-period formula. If $t_p \le 2$ for every prime factor $p|M$, then

$$P_{\min} = \prod_{p|M} p^{e_p},$$

where

$$e_p = a_p t_p - b_p(t_p - 1).$$

This refines the preceding general bounds for the minimal period and is supported by numerical evidence under the stated restriction $t_p \le 2$; its validity beyond these conditions is not asserted here.