We call a subset $A = {a_1, \{a_1, a_2, a_3, \dots}$ dots\}$ of $\mathbb N$ with $a_1 < a_2 < \dots $ transparent if $a_{k+1} - a_k$ goes to $\infty$ as $k \rightarrow \infty$. Is the following true? For every finite set $P$ of prime numbers, the set $A_P := {p_1 \{p_1 \cdots p_s : p_i \in P}$ P\}$ is transparent.
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Subsetes Subsets of $\mathbb N$ with a finite number of prime factors |
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