sum of digits = sum of factors

Question

Assume that some set $A$ is defined as $A=\{x|x\in Z \ni S(10,x)=\sigma_1(x)\}$ where $S(10,x)$ returns the sum of all of x's digits in base 10, and $\sigma_1(x)$ returns the som of all prime factors of x except one. I can only find several elements of $A$ (2, 3, 4, 5, and 7). Is $A$ infinitely big? If so does $\sum_{n \in A} {}^1/_n$ extend to infinity or does it have some definite value. How about in other bases?