Block-counting sequences are not purely morphic

Let m be a positive integer larger than 1, w be a finite word over $\{0,1,\cdots ,m-1\}$ and $a_{m;w}\left(n\right)$ represent the number of occurrences of the word w in the m-expansion of the non-negative integer n (mod m). In this article, we present an efficient algorithm for generating all sequences ${\left(a_{m;w}\right(n\left)\right)}_{n\in N}$; then, assuming that m is a prime number, we prove that all these sequences are m-uniformly but not purely morphic, except for words w satisfying $\left|w\right|=1$ and $w\ne 0$; finally, under the same assumption of m as before, we prove that the power series ${\sum }_{i=0}^{\infty }{a}_{m;w}\left(n\right)t^n$ is algebraic of degree m over $F_m\left(t\right)$.