Large bias for integers with prime factors in arithmetic progressions

We prove an asymptotic formula for the number of integers $\le x$ which can be written as the product of $k(\ge 2)$ distinct primes $p_1\cdots p_k$ with each prime factor in an arithmetic progression $p_j\equiv a_{j}q$, $(a_j, q)=1$ $(q\ge 3,1\le j\le k)$. For any $A>0$, our result is uniform for $2\le k\le A\log \log x$. Moreover, we show that, there are large biases toward certain arithmetic progressions $(a_{1}q,\cdots ,a_{k}q)$, and such biases have connections with Mertens’ theorem and the least prime in arithmetic progressions.