CONO-SUR 2010 Problem 6

Determine if there exists an infinite sequence $a_0, a_1, a_2, a_3,...$ of nonegative integers that satisfies the following conditions: (i) All nonegative integers appear in the sequence exactly once. (ii) The succession $b_n=a_{n}+n,$ , $n\geq0$ , is formed by all prime numbers and each one appears exactly once.