On the Number of Restricted Prime Factors of an Integer

Let ({qgeqq2}) 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_{nleqq 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}, bgeqq 2)}) .