On the density of odd integers of the form (p − 1)2−n and related questions

It is shown that odd integers k such that k · 2ⁿ + 1 is prime for some positive integer n have a positive lower density. More generally, for any primes p₁, …, p_r, the integers k such that k is relatively prime to each of p₁,…, p_r, and such that k · p₁^n₁p₂^n₂ … p_r^n_r + 1 is prime for some n₁,…, n_r, also have a positive lower density.