Carmichael numbers in the sequence (2n k+1)n≥1

A Carmichael number is a positive integer N which is composite and the congruence aN ≡ a (mod N) holds for all integers a. The smallest Carmichael number is N = 561 and was found by Carmichael in 1910 in [6]. It is well– known that there are infinitely many Carmichael numbers (see [1]). Here, we let k be any odd positive integer and study the presence of Carmichael numbers in the sequence of general term 2nk +1. Since it is known [15] that the sequence 2n + 1 does not contain Carmichael numbers, we will assume that k ≥ 3 through the paper. We have the following result. For a positive integer m let τ(m) be the number of positive divisors of m. We also write ω(m) for the number of distinct prime factors of m. For a positive real number x we write log x for its natural logarithm.