Sierpinski and Carmichael numbers

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