We establish several related results on Carmichael, Sierpiński and Riesel numbers. First, we prove that almost all odd natural numbers
k
k
have the property that
2
n
k
+
1
2^nk+1
is not a Carmichael number for any
n
∈
N
n\in \mathbb {N}
; this implies the existence of a set
K
\mathscr {K}
of positive lower density such that for any
k
∈
K
k\in \mathscr {K}
the number
2
n
k
+
1
2^nk+1
is neither prime nor Carmichael for every
n
∈
N
n\in \mathbb {N}
. Next, using a recent result of Matomäki and Wright, we show that there are
≫
x
1
/
5
\gg x^{1/5}
Carmichael numbers up to
x
x
that are also Sierpiński and Riesel. Finally, we show that if
2
n
k
+
1
2^nk+1
is Lehmer, then
n
⩽
150
ω
(
k
)
2
log
k
n\leqslant 150\,\omega (k)^2\log k
, where
ω
(
k
)
\omega (k)
is the number of distinct primes dividing
k
k
.