Since Sum_{prime q} 1/q diverges, the sequence is infinite.
In fact, by the Prime Number Theorem Prime(k) ~ k log k as k > infinity, and by integration Sum_{k <= n} 1/(k log k) ~ log log n, so a(n) ~ Prime(Floor(e^e^n)).
a(4) = A000040(A046024(4)1) = Prime[43922730588128389], but Mathematica 7.0.0 cannot compute this prime on a Mac computer running OS X.
Instead, using a(4) = largest prime < A016088(4) = 1801241230056600523, Mathematica's PrimeQ function finds that a(4) = 1801241230056600467.
See A016088 for other relevant comments, references, links, and programs.
