Subword complexity and non-automaticity of certain completely multiplicative functions

In this article, we prove that for a completely multiplicative function f from N⁎ to a field K such that the set{p|f(p)≠1Kand p is prime} is finite, the asymptotic subword complexity of f is Θ(nt), where t is the number of primes p that f(p)≠0K,1K. This proves in particular that sequences like ((−1)v2(n)+v3(n))n are not k-automatic for k≥2.