On the $P_1$ property of sequences of positive integers

In this paper we introduce the concept of $P_1$ property of sequences, consisting of positive integers and prove two criteria revealing this property. First one deals with rather slow increasing sequences while the second one works for those sequences of positive integers which satisfy certain number theoretic condition. Additionally, we prove the unboundedness of common divisors of distinct terms of sequences of the form $(2^{2^n}+d)^{\infty}_{n=1}$ for integers $d\neq1.$