Given an integer $q\ge 2$, a $q$-normal number is an irrational number $\eta $ such that any preassigned sequence of $\ell $ digits occurs in the $q$-ary expansion of $\eta $ at the expected frequency, namely $1/q^\ell $. In a recent paper we constructed a large family of normal numbers, showing in particular that, if $P(n)$ stands for the largest prime factor of $n$, then the number $0.P(2)P(3)P(4)\ldots ,$ the concatenation of the numbers $P(2), P(3), P(4), \ldots ,$ each represented in base $q$, is a $q$-normal number, thereby answering in the affirmative a question raised by Igor Shparlinski. We also showed that $0.P(2+1)P(3+1)P(5+1) \ldots P(p+1)\ldots ,$ where $p$ runs through the sequence of primes, is a $q$-normal number. Here, we show that, given any fixed integer $k\ge 2$, the numbers $0.P_k(2)P_k(3)P_k(4)\ldots $ and $0. P_k(2+1)P_k(3+1)P_k(5+1) \ldots P_k(p+1)\ldots ,$ where $P_k(n)$ stands for the $k{\rm th}$ largest prime factor of $n$, are $q$-normal numbers. These results are part of more general statements.