  In this article, we prove a theorem \`a la Mauduit et Rivat (prime number
theorem, Moebius randomness principle) for functions that count digital blocks
whose length is a growing function tending to infinity. These sequences are not
automatic. To obtain our results, we control sums of type I and II and use an
adapted and refined version of the carry propagation property as well as
standard methods from harmonic analysis.
