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Sequences with a special property? Or not?

Source: Romanian TST 2 2008, Problem 2

June 7, 2008
number theoryprime numbersnumber theory proposed

Problem Statement

Are there any sequences of positive integers 1≤a1<a2<a3<… 1 \leq a_{1} < a_{2} < a_{3} < \ldots such that for each integer n n, the set \left\{a_{k} \plus{} n\ |\ k \equal{} 1, 2, 3, \ldots\right\} contains finitely many prime numbers?
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