a(n) is the largest k such that no subsequence of n numbers appears more than once in the sequence of the first k prime gaps while overlapping subsequences are allowed.
2, 5, 6, 7, 19, 59, 88, 89, 1213, 1214, 3876, 3877, 3878, 10252040, 105645482
Assumption: The number series is infinitely long.
If not, the number sequence a(n) would end with a finite n and this would mean that there is a function f which can calculate any prime number from nothing but the previous n primes. This is because there would be only a limited amount of possibilities of arranging the input parameters while there is an unlimited amount of primes.