AbstractGiven 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.