Nontriviality
The requirement that the initial terms of a primefree sequence be coprime is necessary for the question to be non-trivial. If we allow the initial terms to share a prime factor p (e.g., set a1 = xp and a2 = yp for some x and y both greater than 1), due to the distributive property of multiplication it is obvious that a3 = (x + y)p and more generally all subsequent value in the sequence will be multiples of p. In this case, all the numbers in the sequence will be composite, but for a trivial reason.
The order of the initial terms is also important. In Paul Hoffman's biography of Paul Erdős, The man who loved only numbers, the Wilf sequence is cited but with the initial terms switched. The resulting sequence appears primefree for the first hundred terms or so, but term 138 is the 45-digit prime 439351292910452432574786963588089477522344721.
Read more about this topic: Primefree Sequence