Let
q
q
be an odd number and
S
q
,
0
(
n
)
S_{q,0}(n)
the difference between the number of
k
>
n
k>n
,
k
≡
0
mod
q
k\equiv 0\bmod \,q
, with an even binary digit sum and the corresponding number of
k
>
n
k>n
,
k
≡
0
mod
q
k\equiv 0\bmod \,q
, with an odd binary digit sum. A remarkable theorem of Newman says that
S
3
,
0
(
n
)
>
0
S_{3,0}(n)>0
for all
n
n
. In this paper it is proved that the same assertion holds if
q
q
is divisible by 3 or
q
=
4
N
+
1
q=4^N+1
. On the other hand, it is shown that the number of primes
q
≤
x
q\le x
with this property is
o
(
x
/
log
x
)
o(x/\log x)
. Finally, analoga for “higher parities” are provided.