Block-counting sequences are not purely morphic

Let m be a positive integer larger than 1, w be a finite word over {0,1,⋯,m−1} and a_m;w(n) 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 (a_m;w(n))_n∈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 |w|=1 and w≠0; finally, under the same assumption of m as before, we prove that the power series ∑_i=0^∞a_m;w(n)tⁿ is algebraic of degree m over F_m(t).