Sierpiński and Carmichael numbers

We establish several related results on Carmichael, Sierpiński and Riesel numbers. First, we prove that almost all odd natural numbers k have the property that 2nk + 1 is not a Carmichael number for any n ∈ N; this implies the existence of a set K of positive lower density such that for any k ∈ K the number 2nk + 1 is neither prime nor Carmichael for every n ∈ N. Next, using a recent result of Matomäki, we show that there are x1/5 Carmichael numbers up to x that are also Sierpiński and Riesel. Finally, we show that if 2nk+ 1 is Lehmer, then n 6 150ω(k)2 log k, where ω(k) is the number of distinct primes dividing k.