On the Number of Restricted Prime Factors of an Integer.

Let $${q\geqq2}$$ be an integer and denote S( n) the sum of the digits in base q of the positive integer n. Our main result is to estimate the sum $${\Sigma_{n\leqq x}\tilde{\omega}(n)}$$ where $${\tilde{\omega}(n)}$$ is the number of distinct prime factors p of n such that $${S(p) \equiv a \,{\rm mod} \,b \,(a \in \mathbb{Z}, b\geqq 2)}$$ . [ABSTRACT FROM AUTHOR]